That means that every slot machine in that bank must be set to a theoretical return of at least 97.4%. In other words the longer your played the closer the ratio of wins to money bet would get to 97.4%. Slot machines are very volatile so in the short run anything can happen. Popular opinion has it that only one machine must be set to 97.4%, or that the overall average must be 97.4%. However I am taking a firm stand that both notions are wrong. My source is an agent, whom I won't name, with the Nevada Gaming Control Board.

Do you have anything on Caribbean Draw Poker?

No, not yet. This game will be a very difficult analysis. However, according to John Grochowski the house edge is 1.5% if the dealer draws to inside straights and 2.4% otherwise.

I have seen a version of Caribbean Stud Poker where you can buy another card for the price of your ante bet. Have you analyzed this variation?

Yes, please see my section on Oasis Poker.

I found another game not listed on your site.

I would be happy to hear about it but I generally wait until a game makes it to Vegas before I analyze it.

Video slot machines often have a bonus round in which the player picks chooses items on the screen to reveal a prize. Is the player's final win predetermined, or does the player actually control his destiny?

The final outcome is not predetermined. It is my understandng that the the prizes are constantly being shuffled while waiting for the player's decision. The moment the player make's his choice his fate is sealed. It may or may not depend on the specific item the player chooses, according to how the game was programmed, but the odds are the same either way.

"Ask the Wizard" questions

Do you know of any website that has a good analysis of backgammon odds/statistics/probability, and are there any particular books you can recommend on any aspect of the game? — Tony

Backgammon is one of my favorite gambling games. I don't write about it because player vs. player games are extremely hard to analyze. I also can't seem to find any new ground to break in the game. So, I'll leave the advice to others. Here are my suggested resources:

Backgammon by Paul Magriel: If there were a Bible to backgammon, this would be it. I'm a proud owner of an old hard-cover edition. This book would be a great place to start. Although it was written in 1976, the advice still holds up well.

501 Essential Backgammon Problems by Bill Robertie: I've been trying to get through this book for years, and I'm still only half way there. It is disheartening to get half the problems wrong, enough to make me think I'm as bad at backgammon as I am at golf. However, with every problem missed, there is a valuable lesson to be learned. For the intermediate to advanced player, this book is a valuable, and humbling, learning tool.

Snowie backgammon software: I play about 1000 games a year against this game. Snowie not only plays a near-perfect game, but tells you exactly how costly your errors are, when you make them. There are lots of other features that I have never explored. If there is one thing I've learned from Snowie, it's that the biggest problem with my game is bone-headed mistakes of not seeing perfectly obvious plays sometimes. Much like chess, one bad move can wipe out 100 good ones.

Motif website: Before I purchased Snowie, I played countless games against Motif. The strategy employed by Motif is very solid, in my opinion. There is nothing like playing against a stronger opponent to improve your own game. December 23, 2008

In the game of chinese poker, 4 players are dealt 13 cards each. What are the chances that one player is dealt the Dragon (a hand with no pairs)? Costa from Ottawa, Canada

The probability that any given player will have a dragon is 4¹³/combin(52,13) = 0.000106. The probability that exactly one player is dealt a Dragon could be closely approximated as 4*0.000106*(1-0.000106)³ = 0.000424, or 1 in 2359. February 14, 2007

Hi. I'm curious about the house edge in a game which, I believe, is played only here in Portugal. The game is called "Banca Francesa" (in a literal translation, French Bank) and is played with three dice. Players can bet in "Big" (sum of 3 dice is 14, 15 or 16), "Small" (5, 6 or 7) or "Aces" (sum is 3 or one spot in each dice) - the dealer will be re-rolling the dice until one of these results shows up. Big and Small both pay even money while the Aces pay 61 to 1. Thank you for your time. -- Carlos from Lisbon

From my Sic Bo appendix we see the probability of a total of 5 or 16 is 6/216, a 6 or 15 is 10/216, a 7 or 14 is 15/216, and a 3 is 1/216. So on any one throw the probability of "big" winning is 31/216, "small" is 31/216, and "aces" is 1/216. The number of ways any of these could win is 2*31+1=63. So given that one of these events did occur, the probability that it was big is 31/63, small is 31/63, and aces is 1/63. The house edge on all three bets is 1.59%. May 2, 2006

In your article on Texas Hold'em Bonus I noticed that you've mentioned the only part of the strategy I have quantified is that the player should fold unsuited 2/3 to 2/7 under either set of rules, and unsuited 3/4 under the Atlantic City rules only. Are these the criteria you've created for your simulation? I would also be interested in how you determine whether to bet on the turn or river. I am trying to figure out what exactly would be "optimum" play. I also would like to know what exactly you mean by "expected" value of the initial hands. I guess in short, my question is how did you determine the optimal strategy for this game? - Benjamin from New Brunswick, NJ

I realize it must be frustrating when I declare the house edge of game under optimal strategy but don't state what the optimal strategy is, as is the case in Texas Hold'em Bonus. The reason for that is I don't know what the optimal strategy is either. The number of combinations in most poker-based games is so vast that it would a very tedious and time consuming task to quantify exactly what the proper strategy is. Instead, I program my computer to cycle through every possible set of cards and make the play with the highest expected value. The expected value is how much the player can expect to win (positive) or lose (negative). That greatly reduces the number of lines of code required. So there is no random simulation. My program looks into the future by looping through all possible combinations of cards and going with the play that results in the greatest win or least loss. Feb. 21, 2006

Firstly, thanks for a great and informative website. I have a quick question about Fortune Bonus Pai Gow. I realize that the fortune bonus bet is a sucker bet as it only pays off less than 20% of the time, however I like to play it anyways, just in case I hit a big hand. At the Trump Casino in Gary, Indiana (near Chicago) the fortune bonus pays out the regular bonuses for a 3 of a kind or better, but also pays 1 to 1 for 3-pair. Statistically speaking, how will this increase your chances of winning on the bonus? What percentage of the time will a player have 3-pair in their hand? Thanks again, -GREG

You're welcome. Here is the number of ways to make a three pair:

No joker: combin(13,3)*10*combin(4,2)³*4 = 2,471,040
Joker used in a pair of aces: combin(12,2)*10*4²*combin(4,2)² = 380,160
Joker used as singleton: combin(13,3)*combin(4,2)³ = 61,776

The total number of possible combinations is combin(53,7) = 154,143,080. So the probability of a three pair is (2,471,040+380,160+61,776)/154,143,080 = 0.0189. So changing a three pair from a loss to a 1 to 1 push would reduce the house edge by 1.89%. Jan. 14, 2006

Would you consider doing an analysis on Texas Hold'em Bonus? This game is all over Atlantic City and is also in the Las Vegas Flamingo. Thank you. -- Roger from Atlantic City

It just so happens that I have four computers cranking away on that game right now. The number of combinations in that game is so huge the game requires about 56 days of computer time to cycle through them all. I should have the analysis completed by about October 20. Oct. 3, 2005

What is the best strategy in one of those booths with the money that blows around while you have to collect as much as possible in a limited amount of time?

I asked this question to Randy Hill of Fun Industries Inc.. He said you should hold your arms straight out, palms down, and let the money blow up against the bottom of your hands and arms. When enough has accumulated you stuff it through the slot. Sept. 18, 2005

What are the odds in Faro? -- Tommy from Houston, Texas

For those who may not know this game, the player picks any one rank, then two cards each are dealt. If the first card matches the player's rank, but the second card doesn't, the player wins even money. If the second card matches, whether or not the first one did, the player loses. If neither card matches, the bet is a draw. Using a single deck, the probability of a win is (3/51)*(48/50) = 5.647%. The probability of a loss is 3/51 = 5.882%. The probability of a tie is (48/51)*(47/50) = 88.471%. The house edge is 0.235%. The house edge per bet resolved (ignoring ties) is 2.041%. Sept. 11, 2005

Tequila Poker looks like fun. I know you get a lot of questions but I just had to ask: what would you do if you got dealt four aces? -Bill from Columbia, Maryland

The expected value of playing High Tequila is 115.904 while Tequila Poker is only 16. So you definitely play High Tequila. Aug. 21, 2005

You are the greatest! I just stumbled across your site a few days ago. My question is in regard to Boston 5 Stud Poker. I saw this game tonight at Mohegan Sun in Connecticut. Their "Ante Bonus" for a Straight is listed at 8 x Ante Bet instead of the 10 x Ante Bet on your pay table. How will this affect the overall odds for this game? Thanks again, keep up the great work!

Thank you for all the kind words. If you lower the bonus on the straight from 10 to 8 the house edge increases from 3.32% to 3.48%. Jan. 9, 2005

How did you arrive at 2072 for the number of straight flushes utilizing 4 cards out of 5 in Four Card Poker?

First I separated the straight flushes into two types, those with four consecutive suited cards and those with five. The number of five card straight flushes is the number of suits * number of spans (ace to 10 as lowest card) = 4*10 = 40. Four the four card straight flushes there are 11 different spans (ace to jack as the low card). In the case of the A234 and JQKA straight flushes, the fifth card can be 1 of 47 (52 less the 4 cards already removed and the fifth card which would form a 5 card straight flush, which were already accounted for). So there are 4*2*47 = 376 straight flushes of span A234 or JQKA. Of the other nine there are 46 possible cards for the fifth card (52 less the 4 cards already removed and two for cards that would form a five-card straight flush). So the number of straight flushes of span 2345 to TJQK are 4*9*46=1,656. So the total number of 4-card straight flushes are 40+376+1,656=2,072. Nov. 8, 2004

In four-card poker, which is more likely a straight or a flush?

Not counting straight flushes and royal flushes the probability of a straight is 1.02% and of a flush is 1.04%. So a flush is slightly more likely. July 28, 2004

I noticed that all video blackjack games that I've played in Vegas pay even money on a blackjack. Is this fair according to the rules of blackjack? Because in a previous question (July 4, 2004) you said, "It is a Nevada state law that an electronic game with representations of cards or dice must be based on fair odds. So the game should be fair with odds the same as in a hand dealt game having the same rules." July 11, 2004

What I meant was that images of cards on the screen had to be statistically fair. For example if you took a tally of each card observed in the initial hand of video poker or video blackjack you would see the distribution approaching a flat line over time, much as you would in a hand dealt game. However, there is no law that the standard rules of blackjack must be followed. The machine can legally offer horrible rules like the player losing on ties. The only caveat is that the theoretical return must be at least 75%.

I was at the Luxor this week and I noticed a blackjack machine that looked much like a video poker machine. Do these things use random generators like online, or do they work like slot machines?

Using a random number generator is the same as working like a slot machine. Online blackjack, slot machines, and video blackjack all use random number generators.
It is a Nevada state law that an electronic game with representations of cards or dice must be based on fair odds. So the game should be fair with odds the same as in a hand dealt game having the same rules. July 4, 2004

In Washington State there are a few clubs with a variation of the payouts for Fortune Paigow. They offer the lower pay-outs (Royal Flush 125-1 instead of 150-1) but they also push on three pair. How does this affect the house advantage on the side bet?

The player loses 0.42% because of the lower royal flush pay-out. However, I show the probability of a three pair to be 1.88%. So, overall these changes decrease the house edge by 1.46%. July 4, 2004

First I want to say many thanks for what I consider to be the best gambling related site on the web. I'm a frequent player at Foxwoods and mostly play Black Jack, Craps, and some Video Poker. However when I look at your House Edge Chart, I see that Catch A Wave has a lower edge than Craps and even Baccarat. This makes me wonder why I only observe $ 5.00 bettors playing the Wave. It seems to me that this would be an excellent game for bigger bettors. Am I missing something?

Thanks. As good as my site is, I would doubt if 1 gambler in 1,000 has visited it. So the vast majority of players don't know what a good bet Catch a Wave is, if played correctly. May 5, 2004

On your Super Fun 21 strategy chart you show doubling your 10 against an ace if you have less that 4 cards. Why is this? Also, do you have a strategy based on simple card counting for this game? Thank you. Keston Hasemann from Oak Lawn, USA

The reason you double with 2-4 cards and hit with 5 or more is because a hand of 6-cards automatically wins, and a 5 card total of 21 pays 2-1. However, these incentives do not apply after doubling. Once the player already has four cards there is a stronger incentive to hit as opposed to double. A nice count strategy has been introduced in the green chip section at bj21.com. June 18, 2002

While at Treasure Island in Las Vegas last week I noticed a game called Triple Shot which consisted of hands of War, Blackjack, and Poker (6 card stud). It looked pretty interesting but I would like to know all the rules and payout schedules. Can you help? Also can you tell me what other Vegas Strip casinos have the game? - Kara from Castaic, California

I've seen it there too, and fortunately took some notes. It is similar to Three Way Action as used to be found at the Las Vegas Club. In Triple Shot, the player may make any combination of three bets. The first is a regular blackjack wager. The second is a poker hand. The third is a war bet. I don't remember if the poker bet is based on the player's or dealer's hand but the best five out of six cards are used. If the blackjack hand doesn't contain six cards then more are added to make six. The following odds table for the poker bet shows the house edge is 3.20%.

Triple Shot
Hand	Combinations	Probability	Pays	Return
Royal flush	376	0.000018	100 to 1	0.001847
Straight flush	1468	0.000072	30 to 1	0.002163
Four of a kind	14664	0.000720	15 to 1	0.010804
Full house	165984	0.008153	7 to 1	0.057071
Flush	205792	0.010108	5 to 1	0.050542
Straight	361620	0.017763	4 to 1	0.071050
Three of a kind	732160	0.035963	3 to 1	0.107890
Two pair	2532816	0.124411	2 to 1	0.248821
Pair	2252472	0.110640	1 to 1	0.110640
Nothing	14091168	0.692151	-1 to 1	-0.692151
Total	20358520	1		-0.031321

Finally there is a war game: the player's first card against the dealer's up card, highest card wins. In the event of a tie, the player loses half. The house edge in the war game is 2.94%. I don't know of any other casinos that have the game. It is probably in trial period and only at the Treasure Island. April 22, 2002

What casinos have super fun 21 could you give me a list? - Moe from Philadelphia, USA

I've seen it at the Regent, New York New York, and Palace Station. I hear it is also at the Sunset Station and Santa Fe Station. Aug. 18, 2001

Just a question about an Oriental dice game, where the players are supposed to guess which side of the die shows up. The players will first place their bets on 1,2,3,4,5,6 (like roulette) and then the "dealer" will roll 3 dice simultaneously. Payouts are 1:1 if the chosen numbers shows up once (on any of the 3 dice), 2:1 if the chosen no shows up twice, and 3:1 if the chosen number appears on all 3 dice. As the player can place any number of bets of the board, what will be the optimum number of bets to place? (assuming all my bets are equal in size) - Jansen Au or Toronto, Canada

The probability of three matching is 1/216. The probability of two matching is 3*5/216. The probability of one matching is 3*25*5/216. The probability of 0 matching is 5*5*5/216. So the expected return is 3*(1/216)+2*(15/216)+1*(75/216)-1*(125/216)=-17/216=-7.87%. There is no optimal number of bets, you will give up an expected 7.87% of total money bet no matter what you do. Feb. 10, 2001

My question has to do with the House edge and element of risk calculations for Casino War for the Casino Niagara Rules (i.e., 3:1 pay out on raise and lose the original wager). How did you come up with these numbers I am currently trying to calculate them? I am having trouble. Thanks for you help. ñ Mark Fallat from Vancouver, Canada

Let's let d be the number of decks. The probability of a tie on the first round is (4*d-1)/(52*d-1)= 0.073955. The probability of a tie in the second round is 12*4*d/(52*d-2)*(4*d-1)/(52*d-3)+(4*d-2)/(52*d-2)*(4*d-3)/(52*d-3) = 0.073974. Lets call p₁ the probability of a tie in the first round and p₂ the probability of a tie in the second round. Then the player return is p₁*(2*p₂ +(1-p₂)/2*(1-2))= -0.023301. Multiply by ñ1 and you have the house edge of 2.33%. I hope I didn't go over this too quickly. Jan. 7, 2001

I have 2 questions here. What is the house advantage on Battle Royale if a second tie wins 7 to 1? I am a Casino-on-net fan. What is the basic strategic for BJ? At some place it says they use 4 or 6 decks. ñ Dany Paradis from Montreal, Canada

For the benefit of those who don't know, Battle Royale is what Starnet calls Casino War. However, Starnet pays 3:1 on the total amount bet. Perhaps what you mean to ask is what if they pay 7 for 1 on the original wager only, which would equate to the same thing. The house edge under these rules is 0.66%. (Update: Starnet is now called World Gaming. Here is my World Gaming review).
Casino on Net follows typical U.S. blackjack rules with the curious exception that if you split tens and draw an ace it pays 3:2 (I only know this because I saw it happen to another player in muli-player mode). However, splitting tens is still a bad idea. The basic strategy is the same as for Cryptologic. Dec. 24, 2000

How do you win money playing solitaire in Vegas? --Pattie from Arlington, USA

I have never seen solitaire played for money in Vegas. I understand in the early days of Vegas people wagered on the standard Klondike variation of solitaire but I don't know anything else about it. Nov. 11, 2000

Hey Wiz, Great Site. I wondered if you knew the House Odds on a game played at Foxwoods in Connecticut, called "Catch A Wave." -- Ken Laird of Boston, USA

Thanks for the compliment. I've been requested to analyze this game quite a few times. However, it is generally my policy to wait until a game goes nationwide before I cover it on my site. Nov. 11, 2000

I SAW YOUR ARTICLE "EXPLORING MICROGAMING" WHICH YOU SAID IT OFFERS THE PAI GOW DOMINO GAME. I WOULD BE VERY MUCH APPRECIATED IF YOU CAN TELL ME WHICH ONLINE CASINO OFFER THIS GAME. THANK YOU. ñ Allan Lam from San Gabriel, California

Three that I know have it are the Caribbean Gold, English Harbour, and Silver Dollar. Also, using all capital letters is rather obnoxious. Nov. 4, 2000

I play webmillion.com and they have 94 numbers of which you have to hit 6 in any order. I would like to know the odds of hitting 6 out of 94 since this site originated nobody has ever won the 3 million. Sandee from Norwood, USA .

The probability of getting 6 out of 94 correct is 1 in 814,216,767. Oct. 15, 2000

I know. I know: CASINO WAR is mindless game. However at Casino Niagara the rules are somewhat different. When there is a tie and you go to war, you place a second wager of equal value to the first and if you win the war you are paid 2:1. If you tie with the dealer after the second cards are dealt you win 3:1. I think these rules reduce the house edge by a good margin. Am I right? - Tibby from St. Catharines, Canada

Thanks for pointing out this variation to me. They use different semantics to explain casino war at the Casino Niagara. What they fail to mention is that the original wager always loses in the event of a tie. Paying 3:1 on a tie after a tie, as opposed to 2:1 decreases the house edge from 2.88% to 1.78%. July 2, 2000

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