Ask The Wizard #164

No. I’m getting lots of people arguing with me about this one. Many writers claim that probabilities can not change if additional information is introduced. So if the probability starts at 1 in 26 then it must stay there. Contrary to what betting system salesmen say, probabilities indeed can change as additional information is introduced. I don’t want to try to teach basic probability here but any college level math book on conditional probability or Bayes’ Theorem should cover this topic nicely.

Let me explain what happened on Let’s Make a Deal. The contestant would choose one of three curtains. One would contain a very valuable prize and the other two smaller prizes. For the sake of argument let’s say behind one curtain was a car and behind the other two a goat. Then Monty would always, I repeat ALWAYS, open up one of the two unchosen curtains to reveal a goat. After hundreds of shows this would imply that Monty Hall (the host) knew where the car was and deliberately opened a curtain that revealed a goat. Obviously when the player chose his curtain the probability it held the car was 1/3 and the probability one of the two unchosen curtains held the car was 2/3. Monty is then predestined to open an unchosen curtain containing a goal. Predestined is the key word here. Because Monty can not open the player’s curtain at this stage the probability of the player’s curtain reveals the car stays at 1/3. The probability an unchosen curtain reveales the car remains at 2/3, however it is now all on one curtain. So after a goat is revelead the probability the player’s curtain has the car is 1/3 and the probability the other unopened curtain has the car is 2/3, making switching a wise choise.

The following table shows all the possible outcomes. In the case where the player chose the curtain with the car I had Monty opening a curtain arbitrarily. You can see that not switching results in a 1/3 probability of winning, and switching results in a 2/3 probability of winnning.

Meanwhile in Deal or No Deal nothing is predestined. Let’s assume on Deal or No Deal the amounts remaining were $0.01, $1, and $1,000,000. With three cases left it IS possible that the opened case will contain the million dollars. The following table shows the possible outcomes with three cases left. Remember, the player can not open his own case.

What the Deal or No Deal table shows is that with three cases left the probability the player opens the million dollar case is 1/3 (hopeless to win), the probability a switching player will win is 1/3, and the probability a switching player will lose is 1/3. Thus the odds are the same to switch cases. Once there are only two cases left the probability each case contains the larger prize is 50/50.

I never play the dragon hand myself because it always loses on copies. Regardless of what is in your hand you have the same odds of making any given hand as the dealer, assuming the same strategy. So if you must play it you may do so whenever you like.

Isn’t there a commandment in the bible about stealing? I show the house edge of that game is 9.3% (ouch!). Here is the appropriate basic strategy for this variation of Ties Lose Blackjack. See column #110 for the version where the player can double on soft hands.

Burning cards has no effect on the basic strategy player. They probably are doing this to discourage card counters. However, they may as well just shuffle earlier. For card counting purposes what is important is the number of cards seen, it doesn’t matter whether the unseen cards are burned or behind the cut card.

As if the normal odds were not bad enough. Shame on the IP. Lowering the 2-number horn bets from 15 to 14 increases the house edge from 11.11% to 16.67%. Lowering the 1-number horn bets from 30 to 29 increases the house edge from 13.89% to 16.67%. No, I don’t know of any other casinos that do this, but I don’t look for this sort of thing either.

Thanks for the kind words. So you have played 680,000/5 = 136,000 hands. According to my blackjack appendix 4 the standard deviation per hand, playing two hands at a time, is 1.91. So the standard deviation of 136,000 such hands would be 136,000^0.5*1.91 = 704 hands. Your losses above expectations are $5,500, or 1,100 hands. So you are 1,100/704 = 1.56 standard deviations south of expectations. The probability of doing this badly or worse can be found in Excel as normsdist(-1.56) = 5.94%.

I say you should have returned the winnings. I have never seen this addressed in any book. However, is a book really required? That is what you have a conscience for. You were asked to make things right, it is the right thing to do so.

Poker is one of my weaker games when it comes to gambling so I turn again to J.B. for this one. Here is what he said.

That move is completely illegal. Anytime someone shows his cards to an opponent, the hand is dead. The only exception is when both players have declared themselves "all-in." At that point both hands are shown and the hand is completed. In this case, the guy that showed the cards to his opponent would forfeit the hand and the other guy would win.

After I got this reply I recalled a scene in a movie, I think Stuey, in which one player offers to turn over one of his cards, chosen by the other player, for $100. I asked J.B. if the scene was realistic. Here is his response to that question.

This always used to happen in tournaments and was perfectly okay back then. It was around 2001 or 2002 that this rule was formally changed in almost all card rooms. I was in a tournament last week and was low on chips. I moved all-in with AQ suited. Everyone folded to the big blind who had lots of chips. After about two minutes of him thinking, he showed me the KQ he had and then went back to thinking about whether to call. I called the supervisor over and said he showed me his cards before making a decision and the hand should be dead. He agreed with me, the hand was automatically declared dead, and he was given a five-minute penalty where he had to leave the table, so it cost him about five hands. The blinds and antes were high, so the penalty cost him several thousand tournament chips.