Ask The Wizard #109

The American Mensa Guide to Casino Gambling has the following "anything but seven" combination of craps bets that shows a net win on any number except 7. Here's how much MENSA advises to bet in the "Anything but 7" system:

  • 5- place $5
  • 6- place $6
  • 8- place $6
  • field- $5
  • total= $22

They claim the house edge is 1.136%. How is that possible if every individual bet made has a higher house edge?


Good question. To confirm their math I made the following table, based on a field bet paying 3 to 1 on a 12. The lower right cell does shows an expected loss of 25 cents over $22 bet. So the house edge is indeed .25/22 = 1.136%.

Mensa Anything but Seven Combo

Number Probability Field Place 5 Place 6 Place 8 Win Return
2 0.027778 10 0.000000 0.000000 0.000000 10 0.277778
3 0.055556 5 0.000000 0.000000 0.000000 5 0.277778
4 0.083333 5 0.000000 0.000000 0.000000 5 0.416667
5 0.111111 -5 7 0.000000 0.000000 2 0.222222
6 0.138889 -5 0.000000 7 0.000000 2 0.277778
7 0.166667 -5 -5 -6 -6 -22 -3.666667
8 0.138889 -5 0.000000 0.000000 7 2 0.277778
9 0.111111 5 0 0.000000 0 5 0.555556
10 0.083333 5 0.000000 0.000000 0.000000 5 0.416667
11 0.055556 5 0 0.000000 0.000000 5 0.277778
12 0.027778 15 0.000000 0.000000 0.000000 15 0.416667
Total 1


The reason the overall house edge appears to be less than the house edge of each individual bet is because the house edge on place bets is generally measured as expected player loss per bet resolved.

However, in this case the player is only keeping the place bets up for one roll. This significantly reduces the house edge on the place bets from 4.00% to 1.11% on the 5 and 9, and from 1.52% to 0.46% on the 6 and 8.

For you purists who think I am inconsistent in measuring the house edge on place bets as per bet resolved (or ignoring ties) then I invite you to visit my craps appendix 2 where all craps bets are measured per roll (including ties).

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."


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%.

First, two columns ago, (May 30, 2004) someone asked about splitting aces and then doubling. You seemed to answer it as if the question meant you could double after you received the second card on top of the ace. I read the question as meaning you could double on the ace only, as if it were two cards that added up to 11. I think that would impact your response, but I'm not sure by how much.


Yes, I interpreted the question that the place could after the second card was dealt to the split aces. If the player could double on each ace alone then that would reduce the house edge by 0.21% (based on infinite decks). With an ace alone the player should opt to double against any dealer up card.

In your June 25 column, you seem to say that you've only seen other counters (I presume, based on them changing their bets and/or strategy similar to how you might, if you were counting) twice in many hundreds of hours of playing. At the same time, if I read you right, you seem to say that you think half the players in a strip type casino are successful counters. That seems to be a contradiction. Maybe you meant one half of one percent....? Assuming you see 10 or 12 players per hour that might be more in line with the math.


What I meant was half of one person, or one person for every two casinos. However you were not the only person I confused so I reworded my original answer to say half on one person, not half of everyone playing.

I’m a big fan of your site. I was playing some hands of Pai Gow Poker on my own and came across an unusual situation. After splitting hands, the Player had a flush: A,Q,10,8,3. The House also had a flush: A,joker,10,5,4. So who wins? The rules simply state that the joker can be used to complete a flush but does it have a rank? Can it be used as a second Ace of the flush suit?


Thanks. I just asked a dealer and he confirmed that the house would win that hand because the joker would be used as a king. The general rule is the joker can substitute for any specific card not already in the same hand as long as it completes a straight, flush, or straight flush. Otherwise it is treated as a fifth-suit ace, thus allowing for the possibility of five aces.

If I roll three dice, what is the probability of getting at least two numbers the same?


The probability all numbers will be different is (5/6)*(4/6)=20/36. So the probability at least two numbers will be the same is 1-(20/36) = 16/36 = 44.44%.

Recently, I was watching an episode of the new a "high-roller" playing, I believe, blackjack. Apparently started to lose more and more, he would tear up the cards! I would have thought this a severe breach of etiquette, if not some actual gaming commission regulation, but when asked to stop, he was insulted that they would ask him! Is this sort of thing generally tolerated and I've just never seen it, or is this guy just used to being allowed to get away with that sort of thing because he's losing tons of money, or something else entirely?


Baccarat (at the big tables) is the only casino game in which players are allowed to damage the cards. An explanation I heard is that Asian players bend the cards anyway as they slowly peak at them that they only use each card once. Therefore as long as the dealer is replacing the cards after one usage the casino may as well let the players do anything with them. Being able to identify cards is of little value to baccarat players anyway because the dealer doesn't take a hole card (as the dealer does in blackjack) and the player has no choice as to whether to hit or stand. However, there are also gaming regulations that stipulate that the tapes must show all the cards in case of a dispute, which isn’t possible if the player tears them up first. In the show you mention the player didn’t know this and I think both parties handled it badly, which led to the hard feelings that the show captured. Had I been the casino manager I would have explained what I just said and then asked the player to lay the card face-up on the table before ripping it into tiny pieces.

On a related note yours truly will be on The Casino sometime this season. The story is some college students try to parlay $1000 into $5000 as quickly as possible. They seek my advice on how to do achieve this goal.

Update: That episode never aired. Probably because of me.

I’ve read quotes similar to this on a couple of different sites: "If the dealer won 40 hands in a shoe and you won 20, this trend is likely to continue until you are broke or until the unfavorable bias is removed through many shuffles". That seems like somewhat "extreme" wording to me but my question is, is there any validity to that concept? Might any clumps of generally favorable (high), or unfavorable, cards make it through one dealer shuffling such that a non-shuffle tracker might take advantage by varying his bets to capitalize on short streaks? By the way, your site kicks ass.


Thanks for the compliment. This theory is called card clumping and would make for good fertilizer if it could be bagged. No legitimate blackjack writer puts any stock in it at all.