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Categories for Craps Questions

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The Dice, and Rolling Odds Bets
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Recently, I was in a casino in Oklahoma, playing craps. There are a few rule changes to the "normal" craps rules. Instead of dice, the casino uses a deck of 54 cards (Aces through 6). The stickman will ask you for a number between 1-3. He'll then burn that number of cards and then put the next two face up. That becomes the dice roll. After approximately half to 3/4 of the deck has been used, a new deck is brought in and the old deck is shuffled.

Also, if you want to make a bet on the table, you'll have to pay a dollar ante to the casino. You pay only $1 per come out roll. Once the point is established you can bet as much/little as you'd like without another payment of ante. The table limits are from 5 dollars to 300 dollars.

If the dealer went through 39 cards (out of 54) before re-shuffling the deck, you can count/see 26 of those cards. Previously, you've said that if there are a lot of 5s and 6s left in the deck, you would bet the "yo 11" bet. Can you develop a more effective strategy and way for betting in this casino? I truly feel that this game is beatable. Would a count of high/low, like counting cards in blackjack, work? Thanks. — Chuck from New York

I still say hop bets, like the yo-11, are the way to go. Using chips, you could keep track of how many cards of each face are left in the decks.

With 26 unseen cards, if any one face had 6 left in the deck, you would have a 43.1% advantage on a hard hop bet (two of the same face), assuming it paid 30 to 1. With only 5 left, the house would have a 4.6% advantage.

The easy hops are even more exploitable. If the two dice sides in majority have at least 10 left combined, both with a minimum of 3 left, out of 26 unseen cards, then make an easy hop bet on those two numbers. If two numbers have 5 left, you will have a 23.1% advantage. If one has 4 and one has 6, you will have an 18.2% advantage. If one has 3 and one has 7, you will have a 3.4% advantage. All this assumes easy hop bets pay 15 to 1.

None of the above takes into consideration the $1 fee. As long as you are making large bets, it won’t make much difference. November 6, 2008

What casinos in Las Vegas have a small table, called a tub, for craps? — Dave P.

According to the the Bone Man at NextShooter.com, here is where and when you can find the tubs:

One Tub at Wild, Wild West (probably open only evenings, weekdays, and on weekends).
One Tub at Ellis Island (probably open only evenings, weekdays, and on weekends).
One Tub at Circus Circus in West Casino section, hardly ever open unless on busy holiday. November 6, 2008

Many Oklahoma casinos are now offering a version of card craps similar to the California game (like CA, OK has some very silly gambling laws). The version I've played uses a 54-card deck, with nine each of ace to six, where the "thrower" calls 1 to 3 burn cards between throws. Suits don't matter. The cards are not returned to the deck, so the odds of the game are not equivalent to a true craps dice game. Obviously, if a 5&4 is the come out roll, this hurts the odds in hitting it again since the same 5 and 4 are not returned to the deck. So, it might make even more sense to play "don't pass" at these tables. In addition, it does allow the player to partially count cards (e.g., laying additional odds on the 4 if very few small cards have been seen). How does not returning the cards to the deck change the odds of playing the pass or the don't pass line? Kristin from Norman, OK
This sounds very promising! If this is true, there would be lots of opportunities to count cards. I don't know if they even allow them, but I think the best opportunities would be on the proposition bets. For example, the “yo” bet, which pays 15 to 1 on an 11, would have a 9.43% house advantage off the top of the deck. However, if no 5 or 6 appears in the first two rolls, the odds swing to a player edge of 5.80%. This same principle would apply to any two-number hop bet. March 4, 2008

I am going on vacation to England and the Rendezvous Casino in Brighton offers different payouts in craps than U.S. Casinos. Could you please tell me the house edge on the various bets that differ?

Place 4,10 - 9 1/2 to 5
Place 5,9 -7 to 5
Place 6,8 - 7 to 6
Any Craps - 7 1/2 to 1
Hardways 4,10 - 7 1/2 to 1
Hardways 6,8 - 9 1/2 to 1
Aces/Midnight - 33 to 1
Ace,Deuce/Eleven - 16 to 1

- Ron L. from Brunswick Ga

I just added a section to my craps section on the Rendezvous Rules. July 25, 2007

Hi - Great website! I've read lots of info here about tipping, but I'm still confused about how to tip for craps in particular. I've never played craps (in fact, have rarely played any table games at all) and am trying to get the rules down before my first attempt.

How exactly do I tip at a craps table? There are several people working the table. Do I put down a bet and they all share any winnings? How do I let them know the bet is for them? When, how often, and where should I place this bet? You said something on one of the pages I read about putting out chips "for the dealers" -- do you just randomly toss out some chips and say "for the dealers"? Do you pass them to a specific person? How much do you tip? What if I happen to think that one person at the table has been particularly helpful, while another has been scowling at my inexperience? Can (or should) you tip one more than another? Thanks for any help and for a great website! — Becky from San Antonio TX

You’re welcome. Thanks for the kind words. In my experience, most players make proposition bets for the dealers. For example, a "two way yo" bet is split 50/50 between the player and dealers. To make this bet, a player will hand or toss the bet to one of the dealers and say "two way yo." However, all the proposition bets are sucker bets, which will cut down the value of the tip by up to 16.7%. As you said, I prefer to give the dealer the tip directly, as opposed to betting it. Before a come out roll I will try to get a dealer’s attention and then put the tip in front of him, saying "for the dealers." I don’t like making pass line tips for the dealers, because I’ve been goaded into tipping extra on the odds, which was more than I intended to offer. If you must make a bet for the dealers, I would put it on the field, saying loudly "dealers in the field."

To answer your second question, dealers are only obligated to share cash tips. Anything else they may keep for themselves. I asked about this at the Venetian and the floorman said dealers may accept personal gifts up to $100 in value. Acceptable gifts can be things with a close to cash value, including gift certificates and unresolved sports tickets. It was quietly added that if a player gave a dealer an envelope, nobody other than the player and dealer would ever have to know what was in it. Should you decide to give a specific dealer a tip I would suggest being discreet about it, putting it in an envelope, and away from the table.June 2, 2007

I was at Foxwoods casino this weekend playing craps. The dealer accepted a $3 commission on a $75 buy bet. What would this make the house edge?– Eric from Boston, MA
Assuming the commission is always payable, then the house edge would be 4% at a $3 commission (3/75), compared to 5.33% at a $4 commission. May 2, 2007
The Grand Victoria Casino in Elgin, Illinois offers a promotion called "Craps for Cash." A shooter wins a $4,000 cash bonus for making all six points on the same hand. All that's required is a bet on the pass line. How does this affect the house edge on this particular game? – John B. from Riverside, Illinois
The probability of making all six points is 1 in 6156. So the value of this to the shooter is $4000/6156 = 64.98 cents. As long as the shooter bets $45 or less on the pass line, and nothing else except the odds, he will have an advantage. April 12, 2007
I am a crap dealer in a casino that offers the fire bet (pay table A, 20.83% edge). The limits on the fire bet are $1-$5 (for players and dealers), but the dealers are limited to $1000 payout. What does that do to the house edge? – Donald from Las Vegas
That is very tight to limit the dealers like that. On a $2 bet the house edge goes up to 29.02%, and a $5 bet it is 41.94%. February 14, 2007
What is the best way to make money at craps consistently? – Tibor from Bradenton
Own the casino dealing the game. February 14, 2007
On a Crapless Craps table in Tunica, you can buy the 2, 3, 11, and 12. You listed the house edge when you place those numbers, but not when one is bought. What is the house edge on buying the 12 for $30 if you only pay the commission of $1 (rounded down from $1.50) when you win? According to my math, it's about .47%, which would make it a VERY good bet. I got this by calculating the total money exchanged on all decision rolls ($211, including the vig) and the amount lost ($1). Am I doing this correctly? I want to make sure because this makes it a VERY appealing bet to make! Please detail how you arrived at the house edge as well, so I can make sure I am, in fact, doing it correctly. Thanks so much! – Will from Rector

I didn’t know they had a buy bet in Crapless Craps. The following table shows the house edge of place and buy bets, assuming there were no rounding of winnings. In your example of a $30 buy bet on 2 or 12 the winnings would be 6*$30-$1=$179. So the expected return is [(1/7)*$179 + (6/7)*-$30] / $30 = -0.0048, so we’re very close. December 13, 2006

Just wanted to know if you knew where the majority of casinos here in Vegas buy their craps tables. And if these companies sold their tables to the public? – Dennis from Henderson, NV
Two suppliers of gaming tables I am aware of are TCS John Huxley and Midest Game Supply. They probably do sell to the public. The price of a craps table at Midwest Game Supply is $3,950. If a used one will suffice, the Gambler’s General Store sells used tables. October 4, 2006

I recently witnessed a situation at the local tribal casino involving protocol at the craps table that puzzled me, and I'm wondering if you clarify it for me. On his come out roll, Player 'A' threw a number and established a point. For the next roll, the player next to 'A' (Player 'B') picked up the dice and shot. It's not clear why he did this, or if he even knew 'A'. One of the other players objected, pointing out that 'B' was not the original shooter. After much discussion and head scratching by the dealers and the boxman, the dice were passed to the next player (Player 'C')who finished the hand (he eventually sevened out).

Was this the correct protocol for this situation, and if so, what is the logic behind it? If Player 'A' for some reason simply didn't want to shoot any more, why shouldn't he be allowed to relinquish the dice? If Player 'B' picked up the dice because he didn't understand the game, or if the stickman mistakenly put them in front of him, shouldn't the dice go back to 'A' to finish the hand? – David from Bainbridge Island, WA

I asked the Bone Man at nextshooter.com this one. Here is what he said.

If a player other than the correct shooter picks up the dice and rolls them, it should be a no-call, no-roll and the dice should be returned to the correct shooter. Though this is indeed the proper ruling, the boxman in some instances may allow for the roll if the result is to the favor of all or most of the players. In some instances, the result of the roll may not effect any of the player(s) wagers. Also... Any player can request that the dice be sent to the next shooter to finish a hand. In such cases the same dice may be sent out or the new shooter can request new dice. Upon the completion of the roll, the SAME shooter can then shoot HIS/HER OWN HAND, thereby having more than one hand. August 9, 2006

Do you have any good rules/setups for playing Craps at home for actual cash. I understand that to keep things legal, I can't take a 'house cut', but assuming a buy-in is enforced (like playing poker at home) is there a good system to play privately for money without becoming "the house" and paying winners out of my own pocket? - Chris K from Los Angeles
You could do a tournament. Every player will buy in for the same amount of non-cashable chips. Establish somebody to be the banker, paying off bets as in normal craps. Whoever has the most chips after some benchmark, for example x 7-outs, wins the pool. Since you will have an even chance with everybody else, I think it would be okay to ask for tips for the use of your house. April 19, 2006
Which casinos in Las Vegas gives the best odds on Craps. - Wayne from Honolulu
The Casino Royale, which offers 100x odds. For complete playing conditions in Vegas please see my new Vegas craps directory. March 27, 2006

In the October issue of Casino Player magazine, Frank Scoblete wrote an article on controlled dice shooting where you state you lost $1800 to Stanford Wong when he rolled only 74 sevens in 500 rolls. Why did you bet on such a small sample (500)? A person who claims to be able to control the dice should be willing to demonstrate their skill with a least 50,000 rolls. Am I wrong in thinking that 500 rolls is such a small sample that just about anything could happen? - Gerald from Alexandria

My webmaster Michael Bluejay had the cover story in that issue. They asked me to write for them again but I have been too busy. Regarding your question, I lost the $1800 to another gambling writer, not Stanford. I would have preferred more rolls but there was an obvious time contraint. Assuming one throw per minute it would take 34.7 days to throw the dice 50,000 times. I wasn't the one who decided on 500 but it seemed like a reasonable compromise between a large sample size and time. You are right that 500 is too few to make a good case for or against influencing the dice, but 500 throws is better than zero. Nov. 9, 2005

I'm walking through a casino and see a craps table with a shooter winning lots of money. I feel lucky and want to place a bet. What can/should I do? Do I have to first make a pass line bet? Can I make a come bet? Can I make either bet with odds? Or, do I wait until the next round of play? -- Todd from Chicago

First, it doesn't make any difference that the shooter is making lots of money. Your odds are the same on an ice-cold table. The past does not matter. However if you are going to play then wait patiently for a come out roll. Never make a pass bet after a point has been established, because you'd lose the ability to win with a 7 or 11 on the come-out roll. However, you could make a come bet right away and still have the chance of winning on a 7 or 11. Sept. 25, 2005

If you have reason to believe that the seven is weighted and is coming up more than it should, does that favor the don't or the pass side of craps? - Haig from Englewood

The fewer the sevens the greater the odds favor the pass line bet. The following table shows the house edge according to the percentage of sevens, assuming the probability of all other numbers is proportional to the fair probability. Sept. 25, 2005

House Edge in Craps According to Seven Probability

Seven Probability

Pass House Edge

Don't Pass House Edge

15.000%

-0.666%

3.499%

15.333%

-0.202%

3.024%

15.667%

0.237%

2.574%

16.000%

0.652%

2.148%

16.333%

1.044%

1.744%

16.667%

1.414%

1.364%

17.000%

1.762%

1.005%

17.333%

2.089%

0.667%

17.667%

2.395%

0.349%

18.000%

2.682%

0.051%

18.333%

2.949%

-0.227%

Hi, if person A makes 1000 consecutive bets on the pass line without backing up his bet, and person B makes 1000 consecutive bets on the pass line and he takes 100X odds whenever possible, doesn't each person lose the same amount of money? - Blake Haas from Thousand Oaks

Yes. I can just imagine the follow up question to be why I recommend taking the odds if doing so doesn't help to win more. What I suggest is betting less on the pass so that your need for action is mostly met by a full odds bet. For example if you are comfortable betting about $90 per bet, and the casino allows 5x odds, then I would drop the pass line bet to $15 and bet $75 on the odds. That will lower the overall house edge from 1.414% to 0.326%. Aug. 28, 2005

I realize that decisions per hour in games like blackjack and craps can depend heavily on factors like the number of other players at the table, the hand shuffle vs. machine shuffle, shooter and dealer speed. Still, I was curious if you could give me a rough approximation of how many decisions per hour an individual can expect at a mostly-full craps table and a blackjack table with both a hand shuffle and machine shuffle. This would help me estimate my expected loss per hour and weigh it against the comps I am being offered.

The following tables show the number of hands/tosses per hour in blackjack, craps, and roulette. The source of the tables is Casino Operations Management by Jim Kilby. Aug. 28, 2005

Hands per Hour in Blackjack

Players

Hands per hour

1

209

2

139

3

105

4

84

5

70

6

60

7

52

Rolls per Hour in Craps

Players

Rolls per hour

1

249

3

216

5

144

7

135

9

123

11

102

In craps 29.6% of total rolls are come out rolls, on average.

Spins per Hour in Roulette

Players

Spins per hour

1

112

2

76

3

60

4

55

5

48

6

35

Thanks for the great site. You recently stated that the average craps shooter lasts approx. 8.5 throws. I normally bet the pass with full odds followed by come bets with full odds. Does it make more sense to quit making come bets after, say four throws given the long term probability that the thrower will seven-out in only three to four more throws? Thanks, Fred

You're welcome, thank you. The dice do not have a memory so after four throws you do not get any closer to sevening out. You could roll 1000 non-sevens and still be no closer or further away from a seven than you were the first throw. There is no optimal number of come bets, just make as many as you find the most fun. Dec. 13, 2004

Is the combined house edge in craps of 0.014% (taken from your chart) on don't pass and laying 100x odds the lowest house edge of any casino game? And, does 0.014% casino edge mean that for every $100 you wager you will lose 1.4 cents?

There are still video poker games that with proper strategy pay over 100%. I've also seen a blackjack game at the Fiesta Rancho and Slots-a-Fun in Las Vegas that had a basic strategy advantage. As I argue in my sports betting section betting NFL underdogs at home against the point spread also has resulted in a historical advantage. So 100x odds in craps is still one of the best bets out there, but not the very best. Yes, 0.014% means that per $100 bet you lose 1.4 cents on average. Dec. 5, 2004

In one of your answers you state that the average number of rolls for a shooter in craps is 8.522551. How is that number obtained? - Steve S., Long Island, NY

First, if the probability of an event is p then the expected number of trials for it to occur is 1/p. Let's call x the expected number of rolls per shooter. The probability that any given round will end in one roll (with a 2, 3, 7, 11, or 12) is 1/3. If the player rolls a 4 or 10 on the come out roll the expected number of additional rolls is 4, because the probability of rolling a 4 or 7 is (6+3)/36 = º. . Likewise If the player rolls a 5 or 9 on the come out roll the expected number of additional rolls is 3.6 and for a 6 or 8 is 36/11. Assuming a point was thrown the probability of it being a 4 or 10 is 3/12, a 5 or 9 is 4/12, and a 6 or 8 is 5/12. So the expected number of throws per round is 1+(2/3)*((3/12)*4 + (4/12)*3.6 + (5/12)*(36/11)) = 3.375758. Next, the probability that the player will seven out is (2/3)*((3/12)*(2/3) + (4/12)*(3/5) + (5/12)*(6/11)) = 0.39596. The probability that player will not seven out is 1 - 0.39596 = 0.60404. So...

x = 3.375758 + 0.60404*x
0.39596*x = 3.375758
x = 8.52551 Nov. 28, 2004

Why is it better to make Come bets with odds than to make Place bets? My math indicates that you make more on a place bet when betting equal units. On 4, to instance, if I were to place $10 I'd win $18. On the come bet I would only win $15 ($5 on the initial bet and $10 on the odds). The other advantage to place bets is that I get to choose which numbers I want to bet and that I win the first time that it is rolled. Am I missing something?

Yes, you are. You are forgetting that a come bet wins on the first roll 22.22% of the time and loses 11.11% of the time. So you are missing the extra value of the first roll of a come bet. However if you had a crystal ball that told you that the first roll would result in a point number then you would be right. Sept. 23, 2004

I'm getting ready to go to Vegas and it has been a few years since I have been there, I know Binion's used to be the best place to play craps, however I understand since the Harrah's takeover that it no longer holds true. Could you please tell me the best places to play craps are?

I don't think the takeover had any effect on the craps at Binion's Horseshoe in Vegas. Although they used to offer 100x odds they ended that long before the federal marshals shut them down earlier this year. The best odds in Vegas can now be found at the Casino Royale (between the Venetian and Harrah's), which offers 100x odds. Sept. 23, 2004

What is your opinion of Card Craps as played by many of the casinos in the San Diego area?

In California dice alone can not be used to determine the outcome of a game. To get around this law many casinos use a hybrid of cards and dice called Card Craps. In this game there is a deck of 7 cards, one each numbered one to six, plus a joker. Using a die two of the seven cards are drawn randomly from the shoe. If the first card is a joker then there is no action on craps bet (although there is joker bet). Otherwise the first simulated die roll is whatever number is on the card. If the second card is the joker then the second simulated die is whatever number was on the first card. Otherwise it is whatever number is on the second card. Mathematically speaking the odds are the same as regular craps. However it is a ridiculous commentary about California gambling laws. May 5, 2004

Mr. Wizard, great site. You are the one true expert on the internet. Because the odds in dice are based on the probability of winning as compared with the payout, I question whether it makes sense to take odds on bets that have a low a probability of winning. For example, the odds on the 4 and 10 are 2 to 1. Is it really a good bet to back up the pass line when the 4 or 10 shows. If someone offered a payout of 10,000 to 1, with the odds of winning 5000 to 1 (and you could only make one bet), it would appear to be a good bet, but the chances of winning are so slim that in reality it is probably a suckers bet. Is there any validity to what I am saying?

Thanks for the compliments. As you know I strictly look at the expected value of a bet. However, if taken to an extreme it doesn't make sense at some point. If you made the proposition to me in your example I would only bet about $200 on it, although it has a 100% player advantage. The reason is if I won I would win $2,000,000 and I don't really need more than that. However in craps a 2 to 1 win is not going to change my lifestyle if I win. 2 to 1 in craps is not a big long shot so I say get aggressive on the odds. If you're uncomfortable with bets that have a low probability of winning you might take up betting the don't pass and laying odds. Feb. 27, 2004

You say the house edge on the pass line bet in craps is 1.414%. Is there any coincidence that this number is the square root of 2?

Just a coincidence I assure you. The exact house edge in craps is 7/495, which by definition must be a rational number. In fact I would argue the house edge in all casino games must be a rational number because there are a limited number of possible outcomes in all games, resulting in a house edge of a perfect fraction. 2 is not a perfect square thus the square root of 2 must be irrational by definition. Therefore the two numbers can not be equal. To be specific the house edge on a $100 pass line bet would be $1.41414141... The square root of 2 is 1.4142135623731... Feb. 27, 2004

Mr. Wizard, you're site is great. I think you may be the only honest expert on the internet. My question is this. I know how odds are calculated in craps but I cannot shake the feeling that once a player gets up multiple bets, either through placing them or betting the pass line and making successive come bets, that the odds shift dramatically into the houses favor. It only takes one seven to wipe out all the bets at one time. In order to win, you must hit each number and after it is hit, a seven would wipe out the remaining numbers.

Thank you for the kind words. I still say that the house edge is not dependent on the number of come bets you make. Yes, it is depressing to establish one come point after another and then lose everything on a seven. However there are other times when the shooter takes ages before rolling a seven and you win lots of come bets along the way. Dec. 2, 2003

Could you provide some details about the mechanics of "buying in", for example, at a crap table? Do dealers take cash only? If so, and I wish to make a substantial buy-in, does this mean I must arrive at the casino carrying nothing but cash (a risky action in LV)? What about travelers' checks? Can I make advance arrangements for credit with the casino Cashier's Office? If so, how does that work exactly? Get specific, please.

I remember my first trip to a casino I didn't know how to actually get chips to play with and purchased them at the cage and walked them to the blackjack table. The proper way to buy in at the table is simply to lay your cash on the table and at the appropriate time the dealer will exchange the cash for chips. However if you wish to play for amounts too uncomfortable or unsafe to carry in cash you can wire funds to the casino in advance. Then all you have to do when you get there is ask the pit boss for chips and he will have you sign something, stating you are buying chips against your cash account. To get off topic a bit I think it is time the United States Treasury should start making $500 bills, making it easier to carry large amount of money. A 500 Euro note already exists, which is worth $598 U.S. dollars at the time of this writing. Dec. 2, 2003

I have a craps question. If I make a $100 pass line bet and then a $100 come bet every roll, what is my average action per roll? For example, I bet $100 on the come-out. The dice show a 4. I bet $100 come bet ($200 total on the layout). A five is rolled. I bet another $100 come bet ($300 on the layout). A seven is rolled. My total action was $100+$200+$300=$600, or average of $200 per roll. What is this number for the long run using this betting pattern? Essentially I am looking for my average bet. Thanks.

Good question. Let's think of this in units as opposed to $100 bets. You will always have a bet on the pass or come. On any given roll the probability there is an old pass or come bet on the 4 is 3/9. This is the probability that by looking back at old rolls you will find a 4 before a 7. Likewise the probability of having a bet on 5 is 4/10 and on 6 is 5/11. So the average overall bet is 1+pr(4)+pr(5)+pr(6)+pr(8)+pr(9)+pr(10) = 1+3/9 + 4/10 + 5/11 + 5/11 + 4/10 + 3/9 = 3.3758 units. This average will not true at the beginning, while you are getting in to the game. It will only apply after all point numbers and the 7 have already been rolled at least once. June 27, 2003

At the craps table, I've observed that 80% of the people bet one the pass line. From what you say, the don't pass line has a slightly lower house edge. So why don't most people bet accordingly? - Dan from Ottawa, Canada

Most player probably don't know the don't pass is a slightly better bet. Even those that do probably bet the pass bet because it feels better to be part of a group than to go against it. May 26, 2003

According to the Las Vegas Advisor the record for the longest time a single shooter held the dice in craps is held by Stanley Fujitake, who once held the dice for three hours and six minutes at a downtown casino before sevening out.

(1) What are the odds that Mr. Fujitake could have accomplished his feat, assuming typical craps speed?

(2) What are the odds that this happened in Vegas since 1950?

- Veggie Boy

I hope you're happy. I think this is the longest answer to any “ask the wizard” question. It has been rewritten many times, with the last rewrite on June 1, 2009.

Yes, this happened at the California casino. There is a plaque in the side of the building mentioning it.

According to The Virgin Kiss by Frank Scoblete, Fujitake had 119 total rolls, including the seven out. The following table shows the probability of being in any of the four possible states in craps before the nth roll. The states are defined as follows:

State 1 = Come out roll
State 2 = Point of 4 or 10
State 3 = Point of 5 or 9
State 4 = Point of 6 or 8

The probabilities are recursive. Let p(x,r) represent the probability of being in state x before roll r. Based on simple dice probabilities, p(x,r) can be expressed as follows:

(1) P(1,r) = (12/36)*p(1,r-1) + (3/36)*p(2,r-1) + (4/36)*p(3,r-1) + (5/36)* p(4,r-1)
(2) P(2,r) = (6/36)*p(1,r-1) + (27/36)*p(2,r-1)
(3) P(3,r) = (8/36)*p(1,r-1) + (26/36)*p(3,r-1)
(4) P(4,r) = (10/36)*p(1,r-1) + (25/36)*p(4,r-1)

In more plain simple English, equation (1) is saying that the probability of being in a come out roll before roll r is the sum of the following:

  1. Product of the probability of being in a come out roll the previous turn, and the probability of staying in a come out roll (12/36).
  2. Product of the probability of rolling for a 4 or 10 the previous roll, and the probability of making the point (3/36), resulting in a come out roll.
  3. Product of the probability of rolling for a 5 or 9 the previous roll, and the probability of making the point (4/36), resulting in a come out roll.
  4. Product of the probability of rolling for a 6 or 8 the previous roll, and the probability of making the point (5/36), resulting in a come out roll.

Equation (2) is saying the probability of rolling for a 4 or 10 before roll r is the sum of:

  1. Product of the probability of being in a come out roll the previous turn, and the probability of rolling a 4 or 10 (6/36).
  2. Product of the probability of rolling for a point of 4 or 10 the previous turn, and the probability of not rolling the desired point or a 7 (27/36).

Equation (3) is saying the probability of rolling for a 5 or 9 before roll r is the sum of:

  1. Product of the probability of being in a come out roll the previous turn, and the probability of rolling a 5 or 9 (8/36).
  2. Product of the probability of rolling for a point of 5 or 9 the previous turn, and the probability of not rolling the desired point or a 7 (26/36).

Equation (4) is saying the probability of rolling for a 6 or 8 before roll r is the sum of:

  1. Product of the probability of being in a come out roll the previous turn, and the probability of rolling a 6 or 8 (10/36).
  2. Product of the probability of rolling for a point of 6 or 8 the previous turn, and the probability of not rolling the desired point or a 7 (25/36).

That said, we can use a spreadsheet to recursively calculate the probability of each state for roll r, based on the probabilities for the previous state, r-1. We know that for roll 1 the probability is 1 for state 1, and 0 for the rest, because a shooter always starts with a come out roll. The following table shows the probabilities of each state for rolls 1 to 200.

Probability States in Craps — Recursive Table
Roll State 1 State 2 State 3 State 4 Total Inverse of Total
1 1 0 0 0 1 1
2 0.333333333333333 0.166666666666667 0.222222222222222 0.277777777777778 1 1
3 0.188271604938272 0.180555555555556 0.234567901234568 0.285493827160494 0.888888888888889 1.13
4 0.143518518518518 0.166795267489712 0.211248285322359 0.250557270233196 0.772119341563786 1.3
5 0.120010764365188 0.149016203703704 0.184461210181375 0.213864359472641 0.667352537722908 1.5
6 0.10262067837728 0.131763946838642 0.159891043878813 0.18185323973522 0.576128908829955 1.74
7 0.0882102876454065 0.115926406525195 0.138281460218538 0.154792716032036 0.497210870421176 2.01
8 0.0759275581209665 0.101646519501464 0.11947222963459 0.131997799368193 0.429044106625214 2.33
9 0.0653874492037588 0.0888894826462592 0.103158289874085 0.112756126817069 0.370191348541173 2.7
10 0.0563258786657103 0.0775650201853209 0.089033753621008 0.0964660461795646 0.319390698651604 3.13
11 0.0485297458312873 0.0675614115832757 0.0768190173186636 0.0826363872540616 0.275546561987288 3.63
12 0.041819421951951 0.058759349659338 0.0662647893593209 0.0708668649906781 0.237710425961288 4.21
13 0.0360417942997806 0.051039415903162 0.0571511083043875 0.0608296067857351 0.205061925293065 4.88
14 0.0310658957348077 0.0442865276440016 0.0492850880642311 0.0522543920178106 0.176891903460851 5.65
15 0.0267795178915465 0.0383925450221358 0.0424983182096798 0.0449171877164818 0.152587568839844 6.55
16 0.0230864183662804 0.0332576617485263 0.0366442337940013 0.038631246439653 0.131619560348461 7.6
17 0.0199039770282004 0.0287909827057748 0.031595595154841 0.0332401484626147 0.113530703351431 8.81
18 0.0171612165385602 0.0249105665340311 0.0272421469514297 0.0286123189402048 0.0979262489642258 10.2
19 0.0147971244603211 0.0215431276569501 0.0234884875846015 0.0246366705247422 0.0844654102266148 11.8
20 0.0127592272071339 0.0186235331527661 0.0202521575800613 0.0212191113256046 0.0728540292655659 13.7
21 0.0110023753581159 0.0160941877324302 0.017461942076074 0.0182797237558737 0.0628382289224939 15.9
22 0.0094877070716207 0.0139043700256753 0.0150563749123014 0.0157504679855001 0.0541989199950976 18.5
23 0.0081817620698106 0.0120095620311933 0.0129824278970223 0.0135732991764919 0.0467470511745181 21.4
24 0.0070557232889404 0.01037079853503 0.011194367274474 0.0116986138919557 0.0403195029904002 24.8
25 0.0060847670452792 0.0089540527827626 0.0096527593179958 0.010083960560786 0.0347755397068235 28.8
26 0.0052475056379875 0.0077296674279518 0.0083236077397257 0.0086929634575678 0.0299937442632328 33.3
27 0.0045255088385124 0.0066718348439618 0.007177606842688 0.0074944206338631 0.0258693711590253 38.7
28 0.0039028928092806 0.0057581276060567 0.0061894957949441 0.0064615445619917 0.0223120607722731 44.8
29 0.0033659667366465 0.0049690778394227 0.0053375009206331 0.0055713206150722 0.0192438661117744 52
30 0.0029029289198866 0.0042878028356747 0.0046028543841564 0.0048039634095353 0.016597549549253 60.2
31 0.0025036052813879 0.0036996736134038 0.0039693790374211 0.0041424548454791 0.014315112777692 69.9
32 0.0021592242942763 0.0031920227569509 0.0034231304784459 0.0035721506653016 0.0123465281949746 81
33 0.0018622231956239 0.0027538877834259 0.0029520885220501 0.0030804447104251 0.0106486442115249 93.9
34 0.0016060810927247 0.0023757863701734 0.0025458913093971 0.0026564819365796 0.0091842407088747 109
35 0.0013851751984362 0.0020495199597508 0.0021956061885034 0.0022909127594927 0.0079212141061831 126
36 0.001194656966999 0.0017680025028858 0.0018935322913494 0.0019756825269911 0.0068318742882253 146
37 0.0010303453586944 0.0015251113716642 0.0016330304253077 0.0017038509123547 0.0058923380680209 170
38 0.0008886348522869 0.0013155577551972 0.0014083764979876 0.0014694368443281 0.0050820059497998 197
39 0.0007664161585174 0.0011347741251124 0.0012146352157215 0.0012672852675297 0.004383110766881 228
40 0.0006610078743911 0.0009788166202539 0.0010475512465805 0.0010929525909283 0.0037803283321537 265
41 0.0005700975637338 0.0008442804442556 0.0009034554279506 0.0009426081532533 0.0032604415891933 307
42 0.0004916909604345 0.000728226593814 0.0007791839343496 0.0008129494296853 0.0028120509182834 356
43 0.0004240681720134 0.0006281184387662 0.0006720086104602 0.0007011290374021 0.0024253242586419 412
44 0.0003657459169253 0.0005417668577436 0.0005795769235575 0.0006046918793107 0.0020917815775371 478
45 0.0003154449629755 0.0004672827961286 0.0004998602041083 0.0005215210042228 0.0018041089674352 554
46 0.00027206204949 0.0004030362575923 0.0004311090280728 0.0004497909648702 0.0015559983000252 643
47 0.0002346456750916 0.0003476208681092 0.0003718147534948 0.0003879276282404 0.001342008924936 745
48 0.0002023752183502 0.0002998232635972 0.0003206763608777 0.0003345735404702 0.0011574483832953 864
49 0.0001745429321349 0.0002585966507563 0.0002765718647117 0.0002885580748682 0.0009982695224711 1002
50 0.0001505384158632 0.0002230379767564 0.0002385336649884 0.0002488716998071 0.0008609817574151 1161
51 0.0001298352244338 0.0001923682185445 0.0002057272949057 0.0002146437959391 0.0007425745338231 1347
52 0.000111979319671 0.0001659153679807 0.000177433096195 0.0001851235317449 0.0006404513155915 1561
53 0.0000965791106527 0.0001430997459307 0.0001530304182899 0.0001596633747315 0.0005523726496048 1810
54 0.0000832968642345 0.0001234213278901 0.0001319839933544 0.0001377048743004 0.0004764070597795 2099
55 0.0000718412972057 0.0001064488066233 0.0001138321872525 0.0001187664027737 0.0004108886938553 2434
56 0.0000619611874782 0.0000918101545018 0.0000981768679503 0.0001024325844833 0.0003543807944137 2822
57 0.0000534398640962 0.000079184480456 0.0000846746685148 0.0000883451801907 0.0003056441932578 3272
58 0.0000460904551538 0.0000682950043581 0.0000730294526154 0.0000761952262703 0.0002636101383975 3793
59 0.0000397517893537 0.0000589029957942 0.0000629858169231 0.000065716255786 0.0002273568578569 4398
60 0.0000342848612847 0.0000508025450712 0.0000543234876342 0.0000566784524496 0.0001960893464397 5100
61 0.0000295697828726 0.0000438160523509 0.0000468524880213 0.0000488836090024 0.0001691219322472 5913
62 0.0000255031541284 0.0000377903364086 0.0000404089708759 0.0000421607792719 0.0001458632406848 6856
63 0.0000219957955175 0.0000325932779945 0.0000348516243278 0.0000363625284189 0.0001258032262587 7949
64 0.0000189707922111 0.0000281109244155 0.0000300585721295 0.0000313616990457 0.0001085019878018 9216
65 0.0000163618073202 0.0000242449920134 0.0000259247003627 0.0000270486221737 0.00009358012187 10686
66 0.0000141116271167 0.0000209107118968 0.0000223593518886 0.0000233287118763 0.0000807104027784 12390
67 0.0000121709063341 0.0000180349717754 0.0000192843379455 0.0000201203907798 0.0000696106068348 14366
68 0.0000104970860282 0.0000155547132205 0.0000166322232571 0.0000173533009121 0.000060037323418 16656
69 0.0000090534602664 0.0000134155492534 0.0000143448470253 0.0000149667606413 0.0000517806171864 19312
70 0.000007808371174 0.0000115705719845 0.0000123720473552 0.0000129084338526 0.0000446594243664 22392
71 0.0000067345146868 0.000009979324184 0.0000106705611286 0.0000111331821682 0.0000385175821676 25962
72 0.0000058083417819 0.0000086069122525 0.0000092030751899 0.0000096020750298 0.0000332204042542 30102
73 0.0000050095420569 0.000007423241153 0.0000079374080332 0.0000082815359324 0.0000286517271755 34902
74 0.0000043205983316 0.0000064023545409 0.0000068458040366 0.00000714260608 0.000024711362989 40467
75 0.0000037264025041 0.0000055218656276 0.0000059043247668 0.0000061603093143 0.0000213129022128 46920
76 0.0000032139242381 0.0000047624663047 0.0000050923239991 0.0000053131043861 0.000018381818928 54402
77 0.0000027719252138 0.0000041075037682 0.0000043919949412 0.0000045824125565 0.0000158538364797 63076
78 0.0000023907126783 0.0000035426153618 0.0000037879797273 0.000003952210168 0.0000136735179354 73134
79 0.0000020619268882 0.0000030554136344 0.0000032670326204 0.0000034086772496 0.0000117930503925 84796
80 0.0000017783577858 0.0000026352147072 0.0000028177295343 0.0000029398944478 0.0000101711964751 98317
81 0.0000015337868869 0.0000022728039947 0.000002430217505 0.0000025355816404 0.0000087723900269 113994
82 0.0000013228509125 0.0000019602341438 0.0000020959986174 0.0000021868724966 0.0000075659561702 132171
83 0.0000011409241759 0.0000016906507599 0.0000018077436486 0.0000018861200428 0.0000065254386273 153246
84 0.0000009840171444 0.0000014581420993 0.0000015591313409 0.0000016267289675 0.0000056280195521 177682
85 0.0000008486889509 0.0000012576094319 0.0000013447097783 0.0000014030109897 0.0000048540191508 206015
86 0.000000731971938 0.0000010846552324 0.000001159776829 0.0000012100601181 0.0000041864641175 238865
87 0.0000006313065794 0.0000009354867473 0.0000010002770294 0.0000010436450648 0.0000036107154209 276953
88 0.000000544485351 0.0000008068328237 0.00000086271265 0.000000900116456 0.0000031141472806 321115
89 0.0000004696043212 0.0000006958721763 0.0000007440669919 0.000000776326803 0.0000026858702924 372319
90 0.0000004050213991 0.0000006001715191 0.0000006417382322 0.0000006695614802 0.0000023164926305 431687
91 0.0000003493203243 0.0000005176322058 0.0000005534823675 0.0000005774791943 0.0000019979140919 500522
92 0.0000003012796097 0.0000004464442084 0.0000004773640041 0.0000004980606417 0.000001723148464 580333
93 0.0000002598457546 0.0000003850464246 0.0000004117139163 0.0000004295642261 0.0000014861703216 672870
94 0.0000002241101424 0.0000003320924442 0.0000003550924406 0.0000003704878666 0.0000012817828938 780163
95 0.0000001932891149 0.0000002864210236 0.0000003062579054 0.0000003195360581 0.0000011055041019 904565
96 0.0000001667067878 0.0000002470306202 0.0000002641394016 0.0000002755914611 0.0000009534682707 1048803
97 0.0000001437802285 0.0000002130574298 0.0000002278132985 0.0000002376904002 0.0000008223413569 1216040
98 0.0000001240066729 0.0000001837564437 0.0000001964829886 0.0000002050017303 0.0000007092478355 1409944
99 0.0000001069525003 0.0000001584851116 0.0000001694614191 0.0000001768086107 0.0000006117076417 1634768
100 0.0000000922437241 0.0000001366892504 0.000000146156025 0.0000001524927853 0.0000005275817848 1895441
101 0.0000000795577908 0.0000001178908919 0.0000001260557345 0.0000001315210243 0.0000004550254414 2197679
102 0.0000000686165062 0.0000001016778007 0.0000001087197617 0.000000113433431 0.0000003924474996 2548112
103 0.0000000591799355 0.0000000876944349 0.0000000937679404 0.0000000978333566 0.0000003384756674 2954422
104 0.000000051041141 0.0000000756341487 0.0000000808723871 0.0000000843787019 0.0000002919263787 3425521
105 0.0000000440216443 0.0000000652324684 0.0000000697503109 0.0000000727744155 0.0000002517788391 3971740
106 0.0000000379675127 0.000000056261292 0.0000000601578122 0.0000000627660231 0.00000021715264 4605056
107 0.0000000327459831 0.0000000485238878 0.0000000518845338 0.0000000541340473 0.0000001872884521 5339357
108 0.0000000282425509 0.0000000418505797 0.0000000447490485 0.0000000466891949 0.0000001615313739 6190748
109 0.0000000243584588 0.0000000360950266 0.0000000385948797 0.000000040268205 0.0000001393165701 7177897
110 0.0000000210085314 0.0000000311310131 0.0000000332870706 0.0000000347302698 0.0000001201568849 8322453
111 0.0000000181193069 0.0000000268496817 0.0000000287092246 0.0000000299539461 0.0000001036321593 9649514
112 0.0000000156274266 0.0000000231571458 0.0000000247609526 0.0000000258344923 0.0000000893800172 11188183
113 0.0000000134782452 0.0000000199724304 0.0000000213556717 0.0000000222815714 0.0000000770879188 12972201
114 0.0000000116246327 0.000000017225697 0.0000000184187063 0.0000000192172705 0.0000000664863065 15040691
115 0.0000000100259406 0.0000000148567115 0.0000000158856507 0.0000000165743914 0.0000000573426942 17439013
116 0.0000000086471106 0.0000000128135238 0.0000000137009567 0.0000000142949775 0.0000000494565686 20219761
117 0.0000000074579059 0.0000000110513279 0.0000000118167156 0.0000000123290429 0.0000000426549923 23443915
118 0.0000000064322481 0.0000000095314803 0.000000010191607 0.0000000106334759 0.0000000367888112 27182178
119 0.0000000055476451 0.0000000082206515 0.0000000087899935 0.0000000091710938 0.000000031729384 31516527
120 0.0000000047846983 0.0000000070900962 0.0000000075811387 0.0000000079098277 0.0000000273657609 36542013
121 0.0000000041266767 0.0000000061150219 0.0000000065385331 0.0000000068220188 0.0000000236022505 42368841
122 0.0000000035591503 0.0000000052740458 0.0000000056393132 0.0000000058838121 0.0000000203563215 49124789
123 0.0000000030696738 0.0000000045487261 0.0000000048637596 0.0000000050746335 0.000000017556793 56958011
124 0.000000002647513 0.0000000039231569 0.000000004194865 0.0000000043767382 0.0000000151422731 66040283
125 0.0000000022834105 0.0000000033836198 0.0000000036179609 0.0000000037748218 0.0000000130598131 76570774
126 0.0000000019693816 0.0000000029182833 0.0000000031203964 0.0000000032556847 0.000000011263746 88780411
127 0.00000000169854 0.0000000025169427 0.0000000026912599 0.0000000028079426 0.0000000097146853 102936943
128 0.0000000014649461 0.000000002170797 0.0000000023211411 0.0000000024217768 0.000000008378661 119350812
129 0.0000000012634776 0.0000000018722555 0.0000000020019232 0.0000000020887189 0.0000000072263752 138381964
130 0.0000000010897163 0.0000000016147712 0.0000000017266062 0.0000000018014653 0.000000006232559 160447740
131 0.0000000009398517 0.0000000013926978 0.0000000014891526 0.0000000015537165 0.0000000053754185 186032027
132 0.0000000008105974 0.0000000012011653 0.000000001284355 0.0000000013400397 0.0000000046361574 215695871
133 0.000000000699119 0.0000000010359735 0.0000000011077225 0.0000000011557491 0.000000003998564 250089780
134 0.0000000006029718 0.0000000008935 0.0000000009553816 0.0000000009968032 0.0000000034486565 289967990
135 0.0000000005200473 0.0000000007706203 0.0000000008239915 0.0000000008597166 0.0000000029743757 336205003
136 0.0000000004485272 0.0000000006646398 0.0000000007106711 0.000000000741483 0.000000002565321 389814765
137 0.0000000003868429 0.0000000005732343 0.0000000006129351 0.0000000006395096 0.000000002212522 451972902
138 0.0000000003336419 0.0000000004943996 0.0000000005286405 0.0000000005515603 0.0000000019082422 524042501
139 0.0000000002877573 0.0000000004264067 0.0000000004559385 0.0000000004757063 0.0000000016458088 607604000
140 0.0000000002481832 0.0000000003677645 0.000000000393235 0.0000000004102842 0.0000000014194669 704489846
141 0.0000000002140515 0.0000000003171873 0.0000000003391549 0.0000000003538593 0.0000000012242529 816824679
142 0.0000000001846138 0.0000000002735657 0.0000000002925122 0.0000000003051944 0.000000001055886 947071928
143 0.0000000001592245 0.0000000002359432 0.0000000002522841 0.0000000002632221 0.000000000910674 1098087828
144 0.000000000137327 0.0000000002034948 0.0000000002175884 0.0000000002270222 0.0000000007854324 1273184054
145 0.0000000001184409 0.000000000175509 0.0000000001876643 0.0000000001958007 0.0000000006774148 1476200350
146 0.0000000001021522 0.0000000001513719 0.0000000001618555 0.0000000001688729 0.0000000005842525 1711588726
147 0.0000000000881036 0.0000000001305543 0.0000000001395961 0.0000000001456485 0.0000000005039024 1984511092
148 0.000000000075987 0.0000000001125996 0.000000000120398 0.000000000125618 0.0000000004346026 2300952450
149 0.0000000000655368 0.0000000000971142 0.0000000001038401 0.0000000001083422 0.0000000003748334 2667852146
150 0.0000000000565238 0.0000000000837585 0.0000000000895594 0.0000000000934423 0.0000000003232839 3093256045
151 0.0000000000487503 0.0000000000722395 0.0000000000772426 0.0000000000805916 0.0000000002788239 3586492968
152 0.0000000000420458 0.0000000000623047 0.0000000000666197 0.0000000000695081 0.0000000002404783 4158379268
153 0.0000000000362634 0.0000000000537361 0.0000000000574578 0.0000000000599489 0.0000000002074062 4821456027
154 0.0000000000312763 0.000000000046346 0.0000000000495558 0.0000000000517044 0.0000000001788824 5590264072
155 0.0000000000269749 0.0000000000399722 0.0000000000427406 0.0000000000445937 0.0000000001542814 6481662846
156 0.0000000000232652 0.000000000034475 0.0000000000368626 0.0000000000384609 0.0000000001330637 7515200125
157 0.0000000000200656 0.0000000000297338 0.0000000000317931 0.0000000000331715 0.0000000001147639 8713540685
158 0.0000000000173061 0.0000000000256446 0.0000000000274207 0.0000000000286095 0.0000000000989809 10102963329
159 0.000000000014926 0.0000000000221178 0.0000000000236496 0.000000000024675 0.0000000000853684 11713937159
160 0.0000000000128733 0.000000000019076 0.0000000000203972 0.0000000000212815 0.000000000073628 13581789749
161 0.0000000000111029 0.0000000000164526 0.000000000017592 0.0000000000183547 0.0000000000635022 15747481849
162 0.0000000000095759 0.0000000000141899 0.0000000000151727 0.0000000000158305 0.000000000054769 18258505628
163 0.000000000008259 0.0000000000122384 0.000000000013086 0.0000000000136534 0.0000000000472368 21169926149
164 0.0000000000071232 0.0000000000105553 0.0000000000112863 0.0000000000117757 0.0000000000407405 24545588905
165 0.0000000000061436 0.0000000000091037 0.0000000000097342 0.0000000000101562 0.0000000000351376 28459519907
166 0.0000000000052987 0.0000000000078517 0.0000000000083955 0.0000000000087595 0.0000000000303053 32997549029
167 0.0000000000045699 0.0000000000067719 0.0000000000072409 0.0000000000075548 0.0000000000261375 38259192197
168 0.0000000000039415 0.0000000000058406 0.0000000000062451 0.0000000000065158 0.0000000000225429 44359833703
169 0.0000000000033994 0.0000000000050373 0.0000000000053862 0.0000000000056197 0.0000000000194427 51433256511
170 0.0000000000029319 0.0000000000043446 0.0000000000046455 0.0000000000048469 0.0000000000167688 59634576023
171 0.0000000000025287 0.0000000000037471 0.0000000000040066 0.0000000000041803 0.0000000000144626 69143641656
172 0.0000000000021809 0.0000000000032318 0.0000000000034556 0.0000000000036054 0.0000000000124737 80168980822
173 0.000000000001881 0.0000000000027873 0.0000000000029803 0.0000000000031096 0.0000000000107582 92952371788
174 0.0000000000016223 0.000000000002404 0.0000000000025705 0.0000000000026819 0.0000000000092787 107774145717
175 0.0000000000013992 0.0000000000020734 0.000000000002217 0.0000000000023131 0.0000000000080026 124959334136
176 0.0000000000012068 0.0000000000017882 0.0000000000019121 0.000000000001995 0.000000000006902 144884796662
177 0.0000000000010408 0.0000000000015423 0.0000000000016491 0.0000000000017206 0.0000000000059528 167987485281
178 0.0000000000008977 0.0000000000013302 0.0000000000014223 0.000000000001484 0.0000000000051342 194774026407
179 0.0000000000007742 0.0000000000011473 0.0000000000012267 0.0000000000012799 0.0000000000044281 225831830862
180 0.0000000000006677 0.0000000000009895 0.000000000001058 0.0000000000011039 0.0000000000038191 261841975396
181 0.0000000000005759 0.0000000000008534 0.0000000000009125 0.0000000000009521 0.0000000000032939 303594138248
182 0.0000000000004967 0.000000000000736 0.000000000000787 0.0000000000008211 0.0000000000028409 352003916251
183 0.0000000000004284 0.0000000000006348 0.0000000000006788 0.0000000000007082 0.0000000000024502 408132903260
184 0.0000000000003695 0.0000000000005475 0.0000000000005854 0.0000000000006108 0.0000000000021132 473211970191
185 0.0000000000003187 0.0000000000004722 0.0000000000005049 0.0000000000005268 0.0000000000018226 548668257185
186 0.0000000000002748 0.0000000000004073 0.0000000000004355 0.0000000000004544 0.0000000000015719 636156469840
187 0.000000000000237 0.0000000000003513 0.0000000000003756 0.0000000000003919 0.0000000000013558 737595165784
188 0.0000000000002044 0.000000000000303 0.0000000000003239 0.000000000000338 0.0000000000011693 855208827359
189 0.0000000000001763 0.0000000000002613 0.0000000000002794 0.0000000000002915 0.0000000000010085 991576643015
190 0.0000000000001521 0.0000000000002254 0.000000000000241 0.0000000000002514 0.0000000000008698 1149689067183
191 0.0000000000001312 0.0000000000001944 0.0000000000002078 0.0000000000002168 0.0000000000007502 1333013398925
192 0.0000000000001131 0.0000000000001676 0.0000000000001792 0.000000000000187 0.000000000000647 1545569817470
193 0.0000000000000976 0.0000000000001446 0.0000000000001546 0.0000000000001613 0.000000000000558 1792019542040
194 0.0000000000000841 0.0000000000001247 0.0000000000001333 0.0000000000001391 0.0000000000004813 2077767049249
195 0.0000000000000726 0.0000000000001075 0.000000000000115 0.00000000000012 0.0000000000004151 2409078589640
196 0.0000000000000626 0.0000000000000928 0.0000000000000992 0.0000000000001035 0.000000000000358 2793219602342
197 0.000000000000054 0.00000000000008 0.0000000000000855 0.0000000000000892 0.0000000000003088 3238614041259
198 0.0000000000000466 0.000000000000069 0.0000000000000738 0.000000000000077 0.0000000000002663 3755029106716
199 0.0000000000000402 0.0000000000000595 0.0000000000000636 0.0000000000000664 0.0000000000002297 4353789433582
200 0.0000000000000346 0.0000000000000513 0.0000000000000549 0.0000000000000573 0.0000000000001981 5048025432896

To verify this table, I did a random simulation of over 21 billion new shooters. The following table shows the count for each number of throws. The average rolls per shooter was 8.53.

Total Rolls in Craps Simulation
Rolls Count Probability
2 2337166954 0.111113765998
3 2456226208 0.116774089950
4 2203680193 0.104767528430
5 1918801263 0.091223793049
6 1659906169 0.078915383142
7 1433725096 0.068162265665
8 1237906639 0.058852649948
9 1068528927 0.050800082105
10 922222110 0.043844352477
11 795811607 0.037834534896
12 686746351 0.032649346344
13 592513250 0.028169309214
14 511244058 0.024305603214
15 441056957 0.020968762813
16 380450121 0.018087388086
17 328223382 0.015604420557
18 283126061 0.013460400352
19 244241934 0.01161176828
20 210694371 0.010016847533
21 181697608 0.008638281259
22 156748704 0.0074521586
23 135198981 0.006427640059
24 116623267 0.005544512076
25 100580892 0.004781824284
26 86732484 0.004123442236
27 74831327 0.003557636541
28 64530253 0.003067902111
29 55658630 0.002646126747
30 48000095 0.002282024104
31 41403022 0.001968385566
32 35715570 0.001697992298
33 30804938 0.001464530665
34 26567244 0.001263061900
35 22912522 0.001089308833
36 19764940 0.000939666255
37 17049802 0.000810582961
38 14701128 0.000698922126
39 12682839 0.000602968480
40 10936628 0.000519949986
41 9430433 0.000448342350
42 8135964 0.000386800609
43 7015476 0.000333530284
44 6049888 0.000287624227
45 5220048 0.000248171912
46 4501337 0.000214002900
47 3881712 0.000184544642
48 3347205 0.00015913307
49 2887420 0.000137273937
50 2492557 0.000118501331
51 2150924 0.00010225939
52 1851857 0.000088041124
53 1599229 0.000076030665
54 1378541 0.000065538699
55 1189435 0.000056548208
56 1027756 0.000048861653
57 883491 0.000042002995
58 763197 0.000036283969
59 658592 0.00003131083
60 566426 0.000026929067
61 488581 0.000023228154
62 421952 0.000020060474
63 363167 0.000017265713
64 312455 0.000014854759
65 271042 0.000012885899
66 234287 0.00001113849
67 200619 0.000009537843
68 173758 0.000008260816
69 149132 0.000007090045
70 129798 0.000006170866
71 111607 0.000005306028
72 95569 0.000004543549
73 83244 0.000003957592
74 71524 0.000003400399
75 61671 0.000002931967
76 53346 0.000002536180
77 45909 0.000002182609
78 39773 0.000001890891
79 34436 0.000001637159
80 29357 0.000001395693
81 25364 0.000001205857
82 21759 0.000001034468
83 18926 0.000000899781
84 16250 0.000000772559
85 14085 0.00000066963
86 12153 0.000000577779
87 10348 0.000000491965
88 9106 0.000000432918
89 7727 0.000000367358
90 6824 0.000000324427
91 5846 0.000000277931
92 5008 0.000000238091
93 4244 0.000000201769
94 3730 0.000000177332
95 3144 0.000000149472
96 2868 0.000000136351
97 2378 0.000000113055
98 2076 0.000000098697
99 1739 0.000000082676
100 1552 0.000000073785
101 1346 0.000000063992
102 1150 0.000000054673
103 950 0.000000045165
104 841 0.000000039983
105 778 0.000000036988
106 618 0.000000029381
107 541 0.00000002572
108 441 0.000000020966
109 412 0.000000019587
110 361 0.000000017163
111 298 0.000000014168
112 259 0.000000012313
113 201 0.000000009556
114 188 0.000000008938
115 160 0.000000007607
116 160 0.000000007607
117 123 0.000000005848
118 122 0.000000005800
119 85 0.000000004041
120 62 0.000000002948
121 70 0.000000003328
122 60 0.000000002853
123 47 0.000000002234
124 34 0.000000001616
125 50 0.000000002377
126 36 0.000000001712
127 41 0.000000001949
128 30 0.000000001426
129 17 0.000000000808
130 16 0.000000000761
131 9 0.000000000428
132 14 0.000000000666
133 11 0.000000000523
134 17 0.000000000808
135 14 0.000000000666
136 8 0.00000000038
137 7 0.000000000333
138 7 0.000000000333
139 3 0.000000000143
140 5 0.000000000238
141 8 0.00000000038
142 4 0.00000000019
143 2 0.000000000095
144 2 0.000000000095
145 2 0.000000000095
146 4 0.000000000190
147 2 0.000000000095
148 3 0.000000000143
149 2 0.000000000095
150 2 0.000000000095
151 2 0.000000000095
152 1 0.000000000048
153 1 0.000000000048
155 1 0.000000000048
160 1 0.000000000048
161 1 0.000000000048
165 1 0.000000000048
166 1 0.000000000048
172 1 0.000000000048
Total 21034000000 1

It seems to me that the simulation confirms my recursive results in the first table.

According to the recursive table, the probability of Mr. Fujitake making it to the 119th roll is 1 in 31.5 million.

In 'The Virgin Kiss', Scoblete tells the story of a skilled dice thrower, by the handle of "the Captain," making 148 throws in Atlantic City (including the seven out), in July 2005. The exact date and casino are not given, that I can find. According to the recursive table, the probability of making it to the 148th roll, assuming random dice throws, is 1 in 2.3 billion.

As far as I know, the longest confirmed roll is by Patricia Demauro, set at the Borgata in Atlantic City on May 23, 2009 (source). Patricia held the dice for 4 hours and 20 minutes, for a total of 154 rolls. According to the recursive table, the probability of making it to the 154th roll is 1 in 5.6 billion.

My thanks to BruceZ for his help with this question.

May 5, 2003

I've just starting learning the game of craps. In craps, the Don't Pass is a better bet for the player than the Pass Line bet. But the few times I've played in the casinos, most people seem to be betting the Pass Line and not the Don't pass. I'm either not correct on the odds b/t the two bets or is there some reason most players are taking the Pass Line bet over the Don't pass line bet?

That is a good question. It is obviously more fun to go with the crowd than against it. The question is why does the crowd favor the pass line? Perhaps it is just tradition. Maybe when people first started playing craps in private games the don't pass wasn't even an option. March 7, 2003

The Kelly strategy for betting requires a positive edge to be effective. I play craps and I give the house less than a 1% edge. Once a week I get comps of $62. I gamble only 1 1/2 hours and my total betting doesn't reach $3000. Theoretically I earn approximately $30 per session. Would the Kelly strategy be helpful to me? - Jerry Tatz from Del Ray Beach, USA

Unless bankroll preservation is very important to you then Kelly betting won't help. I would just flat bet. Nice strategy to milk the comp system. Jan. 11, 2003

I have two friends that have a bet on which game (craps or baccarat) have the best odds for the player. Could you help me settle this. They are both casino workers and are sure they are right. - Charline from Las Vegas

It depends on how the games are played. If optimal strategy is compared to optimal strategy then craps is better. By betting only the line bets and taking maximum odds the combined house edge in craps is well under 1%. The best you can do is baccarat is bet on the banker at a house edge of 1.06%. However it wouldn't surprise me if the actual house edge in craps is higher, due to all the sucker bets players make. Sept. 24, 2002

Before I ask my questions I just want to say your site is phenomenal! I have two craps questions I was hoping you could answer:

1) Your preference is to count the come out roll of 12 in the calculation of the house edge on the don't pass. If one was to choose NOT to count it, would the house edge on the pass line combined with full double odds be exactly equal to that of the house edge on the don't pass line combined with full double odds?

2) Does the overall house edge against player x go up if player x places come bets (which will be backed up with full double odds) after betting the pass line with full double odds? i.e. player x with just a pass line with full double odds = house edge .572%, player x with same bet but places two come bets with full double odds = house edge (.572%)x(3)? - Jay Sparks from Hamilton, Ontario

Thanks for your kind words. Here are my answers.

1. If we define the house edge as the expected loss per unresolved bet (not counting ties) then the house edge on the don't pass would be 1.40%, just barely less than the 1.41% on the pass line bet. If the player can bet more money on the don't pass side, which is the case in real but not Internet casinos, then the combined house edge favors the don't side more the greater the multiple of odds allowed.

2. Assuming the player keeps his odds on during a come out roll then the overall house edge does not change if the player adds come bets, backed up with the odds. However if the player keeps the odds off, which is the default rule, then the overall house edge will actually go up slightly by adding come bets. Dec. 4, 2001

Will you be doing a risk of ruin analysis for Craps? The only analysis I have found online to date seemed to be flawed. - Brian E from Raleigh, USA

I wasn't planning on it. There are so many betting patterns in craps that one analysis would only fit a small percentage of craps players. Aug. 6, 2001

Hi, I'm from Germany and it's difficult to play craps here. I finally found a place where it is possible. But they changed the rules:

1. you may only bet the "right way"
2. it's not allowed to make odds-bets after establishing the point
3. if the come out roll is craps (2, 3, 12) the bet is deferred and if the next roll is 11 you get your money back, otherwise it's lost.

I wonder which effect it has on the house edge? - Toter Mann from Munich, Germany

I went to casinos in Berlin and Hamburg last year and didn't see craps at all. Let's consider rule change 3 first. My craps appendix shows the probability of winning the pass line bet as 244/495. Under the German rules it will be marginally higher. The probability of rolling a 2, 3, or 12, and then an 11 is ((1+2+1)/36)*(2/36)=1/162. So the house edge will be 1/162 less. From the craps appendix we see the normal house edge is 7/495 =~ 1.41%. The German house edge is 7/495 - 1/162 = 0.80%. The combined house edge under full double odds is 0.57%, and 0.43% when laying the odds. So clearly the US rules are favorable. July 18, 2001

When reading literature about the formula casinos use to determine comps, the only formula I ever see used as an example is a formula for blackjack. Assuming the casino determines your average bet by your spread, what formula is typically used by casinos to determine expected craps losses, which in turn, determines available comps. - Tim Patton from San Antonio, Texas

I asked my friend Larry Drummond, a craps dealer and webmaster of Next Shooter for help on this question. Larry can be a bit abrasive but is a good source of hard to find information on craps. Here is what he said, "Comps for craps vary from Casino to Casino and from Boxman to Boxman. A player should get to know the Boxman. The Boxman sets the players average bet and tracks the TIME that a player is at the table. It is easier for the Boxman to track action for COMPS. if the player is consistant in their wagering pattern. Now, I ask you... if a player goes $52 or $54 across after a Point is established with a $5 flat bet on the Pass Line. Is that a $57 or $59 average?... Or a $5 average with a whole bunch of other INDIVIDUAL bets? The answer is... IT DEPENDS ON HOW WELL YOU KNOW THE BOXMAN and HOW MANY TIMES TO YOU ATTEND THIS PARTICULAR CASINO." Larry added in another e-mail the following, "In addition to the information I already sent to you... ODDS on Pass Line and Come bets are often NOT included in the AVERAGE for comps. Same with LAYING ODDS on the Don't Side... as in the long run this should be a WASH. But... If a smart boxman wants someone who is spending big bucks on ANY 7, the worst bet on the table... he would probably average the ODDS and the LAYS to keep the sucker coming back to the casino... you can re-word this to make it a little more palitable for your site... In addition... A good boxman will COMP to the MAX if he sees that the PLAYER is 'Betting for the Boys'. June 6, 2001

In craps, does the house edge change if you make a don't pass bet then remove it if the point is 6 or 8? What if you remove it if the point is 6,8,5,or 9? - Jon Moriarty from Danville, New Hampshire

You should never remove a don't pass bet after a point is made! Likewise you should never make or add to a pass line bet once a point is made. However if you must make a don't pass bet after the point is made then it is best to do so on a point of 6 or 8, since these have the least chance of winning. May 1, 2001 

Is there a casino that teaches how to play craps? ñ Marty from Houston, USA

There are lots of them. Many casinos give free gambling lessons in the mornings when things are slow. To name just one I know the Silver Legacy in Reno offers lessons. Dec. 24, 2000

I just finished reading your section on strategy for craps with great interest. I understand that better the pass line and come bets with full odds is a good strategy. My question is "does the house edge change at all when playing a strategy of pass line with full odds and making a maximum of two come bets with maximum odds?" In other words; how does time (more rolls) and having more money at risk affect these odds if at all? Or should a person stick with just the single bet with full odds? This seems to be a favorite strategy for most knowledgeable players I have met at the dice tables. -- Jim Marlow of Yakima, USA

The house edge is the same regardless of how many come bets you make assuming you always take the maximum allowable odds and leave the odds turned on during a come out roll. How many come bets you make should be a matter of personal preference. Nov. 11, 2000

I played craps for the first time the other night and went from $70 to $700 with small bets on the pass odds and field bets. I then lost it all down to $6 because my bets were too large (by the dealers suggestion), and gained it back to $1000 after slowing down. For this being the first time it seems like a very easy game to win if you have patience, was it beginners luck? Chris, Tyler, USA

It was mostly just luck. Given the low house edge in craps it isn't unusual to do well often. I would recommend staying away from the field bet in the future and sticking to the line bets and the odds. Oct. 15, 2000

I'm a floor supervisor at a local casino and was wondering about a strange play. A player was betting both the pass and don't pass at the same time. My question is if he was betting the do side for $10, I would give him an average bet of $10. Now that he is betting both sides, and probably not risking any funds, what would the average bet be? I know if this patron bet both the red and black on roulette, he should get an average bet of both bets on red and black, since the house advantage of 5.26% make both bets a loser over the long run. ñ Ray from Plainfield, USA

I see you have a second question on this topic. I suppose to be fair he should get the same credit as for one bet of $20. However I don't know how much you value craps play. Some bets carry higher house edges than others. For each $20 he bets the casino can only expect to profit 28 cents. So don't let him milk the comp system if you assume the casinos expected revenue to be greater than that. Oct. 5, 2000

Q: Great site. Do programs like Microgaming "learn" to defeat the player? I was under the assumption that the answer is no until I played "Don't Pass" in craps. Out of 39 "hands" 28 were Pass, 10 were Don't Pass and 1 was a 12 on the come out roll. I realize that this can happen, but it made me wonder. I had read elsewhere that the software can "learn". I didn't really believe it. No, I didn't lose a lot, my betting unit is $1. I'm "testing" onliine for myself. - Rosalyn Blanchette from South San Francisco, California

A: No, programs like Microgaming do not learn to defeat the players. Reputable companies like Microgaming do not need to play any dirty tricks to make money but realize that there is more money to be made long-term by offering a fair game. Even if they did want to cheat the players there would be much easier ways to do so.

However there is a lot to be said about computer teaching themselves to play. I'm not an expert but I do know that computers can learn from past experience in games like backgammon and chess to improve their play. Finding the Edge (Edited by Olaf Vancura, Judy A. Collins, and William R. Eadington) has a paper titled 'A Computer Teaches Itself to Play Blackjack' by Olaf Vancura if you are interested in learning more about the topic. June 25, 2000

Q: As a moderate craps player who is of course interested in receiving every "comp" possible, could you give advice on the best betting (craps) strategy. Tried to find it in your GREAT web site. - Ernie from Hernando Beach, Florida

A: Unlike most gambling writers I don't put much emphasis on betting strategies. There is no one right or wrong strategy. They all behave differently in the short run but in the long run you will give the house the same percentage of total money bet. However if getting good comps is a priority for you then bet more when the pit boss is watching. Otherwise just do what is most comfortable for you. May 6, 2000

Q: I made the mistake of making a pass line bet in craps after a point was established. The next roll was a 7 and I thought I won but the dealer said I lost. Can the player make a pass line bet after a point is rolled? What should have happened? -Larry of Detoit

A: Yes, you can make a pass line bet at any time but you never should make one or add to one after a point is thrown. Once a point is thrown the odds of winning drop to 33% to 45% depending on the point. You can not add to a don't pass bet because that would favor the player. In your case when a 7 was thrown all pass line bets lost including yours. What happened is what was supposed to happen. Feb. 12, 2000

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