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Ask the Wizard #164

In your April 5, 2006 column you state that if there are only two cases left in Deal or no Deal and the million dollars is still in play then the probability my case has the million is 50-50. I disagree. Isn’t this just a variation of the Monty Hall problem? That is, the million is more likely to be on the stage than in his case?

Jason from Pasadena, CA

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.

Let’s Make a Deal

Player
Chooses
Car Curtain
Opened
Probability Win by
Switching
1 1 1 0% n/a
1 1 2 5.56% N
1 1 3 5.56% N
1 2 1 0% n/a
1 2 2 0% n/a
1 2 3 11.11% Y
1 3 1 0% n/a
1 3 2 11.11% Y
1 3 3 0% n/a
2 1 1 0% n/a
2 1 2 0% n/a
2 1 3 11.11% Y
2 2 1 5.56% N
2 2 2 0% n/a
2 2 3 5.56% N
2 3 1 11.11% Y
2 3 2 0% n/a
2 3 3 0% n/a
3 1 1 0% n/a
3 1 2 11.11% Y
3 1 3 0% n/a
3 2 1 11.11% Y
3 2 2 0% n/a
3 2 3 0% n/a
3 3 1 5.56% N
3 3 2 5.56% N
3 3 3 0% n/a

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.

Deal or No Deal

Player
Chooses
Million $ Case
Opened
Probability Win by
Switching
1 1 1 0% n/a
1 1 2 5.56% N
1 1 3 5.56% N
1 2 1 0% n/a
1 2 2 5.56% Hopeless
1 2 3 5.56% Y
1 3 1 0% n/a
1 3 2 5.56% Y
1 3 3 5.56% Hopeless
2 1 1 5.56% Hopeless
2 1 2 0% n/a
2 1 3 5.56% Y
2 2 1 5.56% N
2 2 2 0% n/a
2 2 3 5.56% N
2 3 1 5.56% Y
2 3 2 0% n/a
2 3 3 5.56% Hopeless
3 1 1 5.56% Hopeless
3 1 2 5.56% Y
3 1 3 0% n/a
3 2 1 5.56% Y
3 2 2 5.56% Hopeless
3 2 3 0% n/a
3 3 1 5.56% N
3 3 2 5.56% N
3 3 3 0% n/a

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.

Love the site!! As mentioned in response to another question, some casinos offer a Dragon Hand in pai gow poker. My theory is to play it if I have, trips or better, preferably without the joker. Others play when they have nothing. Is there any logic to either methodology?

Michael from Las Vegas

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.

I deal blackjack at my church festival and I’d love to run a basic strategy scenario for our rules...

4 decks
Dealer stands on soft 17
Double on 9,10,11
Split any cards
Double after split
PLAYER LOSES PUSHES

The last rule is what I call’s "God’s House Advantage" and is the one rule that obviously is not in any of the strategy generators I’ve found online. I’m convinced that the push loses rule might make hitting a 17 a basic strategy play in some cases. Best regards and keep up the great work.

Joe from Floral Park, NY

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.

I was playing baccarat online at USACasino which uses a live dealer and Playtech software. Apparently Playtech has instituted a new rule that the dealer burns a card after each hand is dealt. This is not the way it is played in casinos. What bearing, if any, does this have on the odds of the game. I can’t believe the casino would institute a new rule that wasn’t in their favor.

Phil from Yonkers

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.

I recently visited Imperial Palace, Las Vegas and played craps, and was very surprised to find out that horn bets there pay 15 FOR 1 and 30 FOR 1 (14 to 1 and 29 to 1 respectively) How dramatically does this change the house edge, and do you know of any other casinos that do not pay the standard of 15 to 1 and 30 to 1?

Charles from Buffalo

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.

Hi wiz, as always, thanks for the great site and love your columns and especially the relationships questions. I have a question about my online blackjack play. I have been playing for quite a while and keep a record or my results. I play two $5 hands against the dealer, thus a total initial bet of $10. My total amount wagered, including doubles and splits, is just over $680,000 at this moment. I calculate my expected loss to have been $3,000 (after factoring the number of total bets into initial bets/hands) at a house edge of 0.5%. However, my actual loss is a far greater amount of $8,500. For such a reasonable sized sample, that seems to be a fairly large discrepancy. I don’t know if that means the game is less than fair (I have no reason to believe so) or I am just experiencing an unfortunate result. What do you think? Are there any reasons to be suspicious based on these results? PS: fortunately, I am still making money due to the bonuses but not as much as I would have expected, and that is the disappointing part. Thanks and all the best.

Mick from Port Kembla

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 136,0000.5*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 had a blackjack and the dealer paid me. Then at the end of the deal he turned his cards over and he discovered he had a blackjack. He forgot to check his hand before paying me. The supervisors wanted me to return the winnings. I refused. Seems I remember these scenario from a book on blackjack. Was I correct?

Woloshen from Montreal

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.

I made an all-in bet after the flop, the only guy left in the hand at the time thinks for a few seconds, shows me his hand. The cards never touched the table, he didn’t say a word, but I took this as him folding. I proceeded to muck my cards, and grab the chips, and he states that he hasn’t folded yet. Now he has seen my two cards, decides to call, and wins on the river card. Is this legal or not?

Craig M. from Brockville

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.