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Ask the Wizard #44

If I roll a die, my probability of rolling a six is 1/6. If I roll two dice, does my probability of rolling a six on one of them increase, or does it stay at 1/6?

Mike R. from Rosemount

If you rolled x dice the probability of getting at least one 6 is 1-(5/6)2. In the case of two dice this is 30.56%.

First let me say, I think your web site is really great. I have told a few people about it, and hope they will try it too. I wish you continued success with it. I also liked the link to WinPoker. I liked WinPoker enough to order it. This is a great program. I have a question that I am hoping you can help me with. I have been trying to figure out the number of times each hand in seven-card stud occurs. I have a copy of your seven-card table, but I am interested in the mathematics to arrive at those numbers. I can figure out the five-card numbers, but the seven-card just baffles me. I would like to send an Excel 2000 file with my numbers. I would also like to know how to figure the number of straights in a 53-card deck with a joker. H E L P ! ! !

Stan from Harahan, Louisiana

Thanks for your kind words. I agree that calculating the numbers for seven-card stud is hard. That is why I do it my computer. My program goes through all possible combinations and scores each one. The number of wild straights in pai gow poker is 11*(44-4)+10*3*(44-4)=10332. Combined with the 10200 natural straights the total is 20532.

I agree with you that there are no system that can beat a negative expectation game. Anyway, I take a look at the cancellation system and keep wondering ... what about using it in a bet like the banker in baccarat, where you have a POSITIVE expectation outcome? In which extension would the commission payed to the house erode your gains in the long run? I apologize for my shady English.

Marcio from Sau Paulo, Brazil

The Banker is baccarat is not a positive expectation bet. You're confusing the probability of winning the bet with having a positive expectation. Even without a betting system, you will probably win any banker bet but you will win less than what you bet, because of the 5% commission. This makes the banker bet a negative expectation bet.

As a math/statistics instructor, I must say that your site should be required reading before anyone ventures into a casino. I like to bank when playing Pai Gow Poker in Nevada casinos. In Tahoe I can usually bank every other hand. I'm in Vegas far less often and there are more casinos to check out. Do you know which ones will allow you to bank every other hand at a full table (if no other players wish to bank)? It seems to be one in seven some places and one in twelve others.

Also, some Tahoe casinos have a jackpot game based on how a players five-card hand. A sucker bet, but as a banker I like it when other players bet it. They'll often set their hand for the jackpot (paid by the house) at the expense of there standard wager (against me) by splitting two high pair to play a straight with 2 singletons, or keeping a full house together and putting two singletons up instead of 3 down, pair up. Do you have any idea which Vegas or Reno casinos do this?

Tom from Fairfield, USA

Thanks for your kind words. Actually, I have been asked about teaching a coarse on the mathematics of gambling at UNLV. pai gow poker is not my game, so I don't follow the details very closely. I do know, as you stated, that some rotate and some zig-zag the banker between the players and dealer. However I don't keep track of who does it which way, sorry. I've also seen that progressive side bet at lots of casinos around town. Again I don't keep track of who specifically has it. However that is a great idea of banking against it, I have never thought of that. Sorry I wasn't of much help.

What are the odds of winning a standard game of Klondike Solitaire as in the Windows version?

James from Greeley, USA

That is probably the most frequently asked question I get which I don't have an answer to. An exhaustive Klondike Solitaire has never been done. Maybe, when computers are a million times faster, somebody will eventually do it. However, it is rumored that Vegas casinos used to offer the game at least back in the fifties. I've asked a number of Vegas old-timers to verify that, but none have been able to, so far.

First of all, I think your web site is great and want to thank you for all this great, solid information. My question is this: blackjack tables have a maximum bet limit that more than increases when you move up from the five dollar table to the ten etc. I suppose this is to discourage larger stakes gamblers from playing at lower stakes tables but how do they calculate these limits? I notice that they are different at different casinos. Also, on your chart of the house edge, it would be great to see a comparison of house edge for a blackjack player using Basic Strategy vs. one who is counting cards. Thanks again for the great work.

Michael from Santa Cruz, USA

You're welcome! Casinos generally try to keep the maximum bet about 200 to 500 times the minimum bet. Why? If a casino is comfortable with a $10,000 bet on a $100 table, why not take it on a $5 table? The answer seems to be that casinos like to corral their big bettors into certain areas. Such high-limit areas tend to have the best staff and surveillance. Limiting the ratio of maximum to minimum bet is also a defense against cheating and advantage play.

The advantage of card counting depends on how good and aggressive the card counter is. Other than my introduction to card counting, I leave that topic to other gambling writers.

What is the casino average hold for craps?

Mary from Rising Sun, USA

I don't know the hold for any game. For the benefit of other readers, the hold percentage is the ratio of casino profit to chips purchased at the table. Since the same chips will circulate back and forth between the players and dealer for an unknown period of time, the mathematician has no way of calculating the hold or hold percentage.

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! Once a point is made of 6 or 8 the don't pass has equity of 9.09% of the bet amount, which you would be throwing away by taking the bet down. The equity of a don't pass bet on a point of 5 or 9 is 20%, and on a 4 or 10 is 33.33%.

On average, in single-zero roulette, how often will a number repeat (ex. two 8s in a row) over the course of 36 spins?

Jon from Danville, New Hampshire

You can expect a repeat once every 37 pairs of numbers. So, with 36 numbers we have 35 pairs of numbers. So, the expected number of repeats is 35/37 = 0.9459.

Your web site is awesome! I enjoy your advice and am amazed at the simple way you explain things about gaming in a way I can understand. My question is this -- while playing a benefit blackjack game where they have from what I could tell, all the "Vegas" rules, I found out they were paying two to one for a blackjack. This seems like a big benefit to the player; just how big is it? (this is no joke or prank, I actually played at the table where they were doing this!)

Bob from Canton, Ohio

Thanks for the compliment. If I had been at this game I would have played it hard. Assuming six-decks and otherwise Vegas rules, the player edge would have been 1.94%. The 2 to 1 on blackjack adds 2.37% to the player's expected return in a six-deck game.

In your slot machine advice, you indicate to play machines with variable states that are in a high state or progressive machines with the meter high. Could you please explain? How do you know a machine is in a high state?

Ken from Naperville, Illinois

For variable-state slots, you have to know what the positive point is for that model of machine. For example, on the Piggy Bankin' slot machine, I think it becomes positive when there are about 40 credits in the bank. At that point the player is supposed to play one coin at a time until the bank is hit. The book Robbing the One-Armed Bandits by Charles Lund (1999) covers specific positive points for various machines, however many of the machines covered in that book are now hard to find.

As for how to determine when a progressive jackpot is unusually high, you'll either have to observe it over a long period of time or find someone who has done the same. For example, keeps data on progressive slots at online casinos. But even when a progressive slot is unusually high, it's impossible to know at what point it becomes high enough to be a positive-expectation game without knowing how the probabilities on the machine are programmed. In my section Deconstructing Megabucks I attempt to figure out when the jackpot is large enough to have a player advantage.

Update: Since this question was published, is blocked to U.S. traffic.

Great site! Is it better to go max three quarters on a quarter machine or drop one dollar in a dollar machine?

"Anonymous" .

Thanks. This is a good question and I wish I had a firm answer. The exact answer depends on the theoretical return of both machines, and nobody ever reveals this information. Yes, you do get a better return in general on dollar machines than quarters, but you are giving up the max-coin bonus. I think the house edge will do down about 2% making the jump from quarters to dollars. However, without reel weightings, I can't tell you the cost of not playing max coins. My general advice is to find a slot machine without a max-coin incentive and then bet one coin at a time.

How would I go about getting a free room at Las Vegas on my first visit?

Dan from San Lorenzo, USA

First, you need to get a player card. Then you have to present it to the pit boss when you play a table game. As a rule of thumb, to get a free room you probably need to bet at least $50-$100 a hand at least four hours a day for every day you stay there. The better the place the more difficult they will be to impress.

I just came back from Vegas and played at the Las Vegas Club. They have a game called "Most liberal 21" with the following rules:
  • Eight decks
  • Dealer hits soft 17
  • Double any first two to four cards
  • Double after split allowed
  • Late surrender allowed
  • Re-split aces allowed
  • Six-card Charlie

What kind of advantage does the house have with these kind of rules?


At the time you asked this they still paid 3-2 on blackjacks. Before considering the rule about doubling on 3 or 4 cards, and the six-card Charlie, the my blackjack house edge calculator says the house edge is 0.50%. My list of rule variations says that doubling on 3 or 4 cards is worth 0.23%, and the six-card Charlie rule is worth 0.16%. So, the overall house edge is 0.50% - 0.23% - 0.16% = 0.11%.

However, since you wrote they changed their rules to pay 2 to 1 on suited blackjacks and 1 to 1 on all other blackjacks. This increases the house edge by 1.13%, to 1.24%. They still have the sign stating it is the "World's most liberal blackjack," which is patently untrue, if you interpret the word "liberal" to mean lowest house edge.