Q: Can you calculate what the probability is of two numbers coming up behind each other in a roll of the dice? Meaning what is the probability of two 4&rsquo;s or two 6&rsquo;s or two 7&rsquo;s back to back? I realize that the past cannot predict the future but is there a way to calculate 7/36 X 7/36 happening back to back? I hope that makes sence.

Sure. That would be Pr(2)2 + Pr(3)2 + ... + Pr(12)2 = (1/36)2 + (2/36)2 + (3/36)2 + (4/36)2 + (5/36)2 + (6/36)2 + (5/36)2 + (4/36)2 + (3/36)2 + (2/36)2 + (1/36)2 = 11.27%.

Question 1

What is the probability that any two chosen ranks, for example queen and king, will appear consecutively in a random deck? Somebody challenged me to an even-money bet that it would happen.

Accepted Answer

According to a random simulation, the probability is 48.64%. So, I would have taken that bet.

Question 2

Great site. I notice you have a lot of information on historical betting spreads. I wanted to run some analysees on historical NBA spreads to test a theory. Any advice on where I can get the data?

Accepted Answer

Thanks. I get a lot of my data from Davler Sports. For college football, I use the free data at The Gold Sheet.

Question 3

Can you calculate what the probability is of two numbers coming up behind each other in a roll of the dice? Meaning what is the probability of two 4&rsquo;s or two 6&rsquo;s or two 7&rsquo;s back to back? I realize that the past cannot predict the future but is there a way to calculate 7/36 X 7/36 happening back to back? I hope that makes sence.

Accepted Answer

Sure. That would be Pr(2)² + Pr(3)² + ... + Pr(12)² = (1/36)² + (2/36)² + (3/36)² + (4/36)² + (5/36)² + (6/36)² + (5/36)² + (4/36)² + (3/36)² + (2/36)² + (1/36)² = 11.27%.

Question 4

This question is about counting cards in a two-deck blackjack game. Using the Knock-Out Count, how do you account for the card the dealer burns and doesn&rsquo;t show you? I usually just assume it is a negative, but there has to be a better way than this conservative approach.

Accepted Answer

A burned card is just like any unseen card left in the shoe. You don&rsquo;t have to assume it is anything. In the Knock-Out count you can completely ignore it, because there is no true count conversion. In the Hi-Lo, or any count with a true count conversion, a perfectionist would add the number of burn cards to the number of cards remaining in the deck. However, since there is usually just one burn card, it can be safely ignored under any card counting strategy.

Question 5

Recently casinos have started an option for the player called "Automatic Win," which means if the player has 20, and the dealer has a 10 showing, the player could win half of his wager right away, without taking the chance of the dealer having a 20 or pulling a 21 somehow. The person who came up with this says that it happens more than half of the time that the player&rsquo;s 20 will either be a push or lose. I don&rsquo;t know if I can agree with his math, please let me know, thank you! P.S. Keep up the good work!

Accepted Answer

That is not true about pushing or losing more than half the time. From my blackjack appendix 2 you can see that when the dealer has a 20 the possible outcomes are as follows, after the dealer peeks for blackjack, and based on six decks.

Dealer gets 17-19 or busts: 59.4%
Dealer gets 20: 36.8%
Dealer gets 21: 3.8%

So, the player will push or lose only 40.6% of the time. The value of a 20 against a dealer 10 is prob(win)-prob(loss) = 59.4% - 3.8% = 55.6%. That is more than the 50% you get by invoking the automatic half win, so you should decline the option. I address this option in my blackjack appendix 8, under the title casino surrender.

For the same reason, you should also decline “even money” when you have a blackjack against a dealer ace. In both cases two birds in the bush ARE worth more than a bird in hand.

Question 6

This is a follow up on Deal or No Deal, which I watched for the first time recently. Your analysis assumes that the house doesn’t know the value of the money in the suitcase. However, in the show I watched, in the endgame both contestants had selected a valuable case, and both were offered (or would have been offered, as one had already quit) above expected value (EV) deals. In the most extreme case, a player "would have been" offered $687K when the two dollar amounts left were $500K and $750K. The only rational explanation for this is that the banker knows the value of the player’s suitcase and the deals offered are based on that.

Just my two cents, and no reply is necessary.

Accepted Answer

Thanks for not expecting a reply, but I usually do reply to game show questions. They claim in every episode that the amounts in the cases are randomly placed, and that neither Howie, nor the banker, know the results. This was never claimed on Let’s Make a Deal, where Monty Hall obviously did know. I too have seen the banker offer more than expected value as the last offer, especially when large amounts are involved. In my strong opinion, this is not because the banker knows what is inside the player’s case. In the 1950s there was a huge scandal when it became known that the show 21, as well as others, were fixed. There is no compelling reason to ruin a successful show, and the integrity of all game shows, to skim some prize money via the bank offers.

I can offer three theories why the banker sometimes offers more than the average of the remaining cases.

The show tries to portray the banker as sweating the money in his office. Howie Mandel is often commenting on the banker’s mood and tone of voice. Maybe it makes the show more dramatic to think of the banker as a risk-averse bean counter, preferring to cut his losses, than risk giving out a big prize.

The real banker truly is risk-averse. This is getting out of my area of expertise, but from my understanding, game and reality shows are usually produced by a company independent from the television network. These smaller companies will seek out an insurance company to mitigate the risk of contestants winning the larger prizes. In such a case, the insurance company would be the real banker, and may be influencing the behavior of the banker on the show. The insurance companies that insure odd-ball stuff like this are not gigantic, and may prefer playing it safe when large amounts are involved.
In your example, the banker offer was 9.92% above expected value. If the banker were following the Kelly Criterion, such an offer would have been made with a total bankroll of only $782,008, which is less than the maximum prize. No self-respecting insurance company would be that conservative. Clearly, this reason alone cannot justify the offer in your example.

The show is trying to make the contestants look stupid and greedy. Shows like Are You Smarter than a Fifth Grader and the Tonight Show's “Jaywalking” would not be successful if we didn’t find some satisfaction in laughing at the trivia-challenged. The shows Friend or Foe and The Weakest Link were outstanding at exposing greed in human nature. I must confess a sense of schadenfreude when a contestant refuses an above expected value offer, and walks with the lower amount in his case.

I tend to think the reason is a combination of these three reasons, but mainly the third.

If I ended this answer here, I’m sure I would get comments, questioning whether the hypothetical banker offers would have really been made. The implication being that they are puffed up for dramatic effect. I have recorded the specifics of 13 games. In one of them, with three cases left ($1,000; $5,000; and $50,000), the average was $18,667, and the offer was $21,000. That is 12.5% over the expected value. In another show, with two cases left ($400 and $750,000), the average was $375,200, and the offer was $400,000. That is 6.6% above expected value. So, I see no reason to question the integrity of the hypothetical offers.

Links:

Deal or no deal formula: This page shows old, and new, formulas for calculating the banker offer, based on the free game at the Deal or No Deal web site.

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