GT-NO: Some Probability Explanations For Gamblers

Last issue, we looked at some probability questions and answers based on my new book with Justin Conrad from the University of Georgia called Probability and Statistics The Vegas Way. I promised some more extensive explanations of each answer, and here I deliver.

Question: What is the probability of throwing three coins and obtaining exactly two heads?

Answer: To get exactly two heads when you flip three coins, you need it to come HHT, HTH, or THH. Each of these outcomes has a probability of one-half cubed or one-eighth. Thus, the total chances are 3/8.

Question: If an urn contains six white balls and four black balls, what’s the probability  that grabbing five results in three black ones and two white ones?

Answer: With six white and four black to choose from, one way to choose three black and two white (picked consecutively, but it’s not necessary) would be BBBWW. The chances of that would be 4/10 × 3/9 × 2/8 × 6/7 × 5/6 = 1/42. But those two white balls have 10 different ways they could be arranged in different positions. Thus, the answer is 10/42 or 5/21.

Question: You have $100 to your name for the next few days and are contemplating betting it all on a 60% shot getting even money. But if you wait until tomorrow, you can bet that $100, but no more, on an 80% shot. Should you wait? What if the second bet was 70%?

Answer: If you play the 60% and then go on to the 70% if you win the first, your EV is 42% of $200 minus 40% of $100. If you wait for the 70% shot, your EV is 70% of $100 minus 30% of $100. Since the first alternative is $44 and the second is $40, you shouldn’t wait even though the second bet is somewhat  better. But if it’s 80%, the EVs are $56 if you play both when you can, versus $60 if you wait.

Question: If you are x standard deviations above the mean what are the chances you are x plus one standard deviations above it?

Answer: You are in the top 16% if you are one standard deviation above the mean, about the top 2 1/4% if you are two standard deviations above and about one eighth of one percent if you are three standard deviations. In eighths that is 128, 17 and one. Someone who is above one SD is only about 17/145  to be two standard deviations or better and someone who is two standard deviations above the mean is even less likely to be that extra third standard deviation as it’s about 1/18.

Question: If a baseball team is 40% to win each game of the World Series, what are the chances they win in exactly six games?

Answer: The key to this one is that for a team to win in six, the result after five must be 3-2. WWWLL is 4/10 × 4/10 × 4/10 × 6/10 × 6/10.  But there are ten ways those wins could be distributed among the five games, so you multiply by 10. 23,040/100,000. Since the 40% team then wins the sixth game, the answer is 40% of that number. (I believe there is a misprint in the answer in the previous column).

Question: A draw poker machine allows you to draw twice. The second draw is to the results of the first draw. You always go for it regardless of what that first draw brings. What’s your chance of making the Royal?

Answer: If you are playing a double draw poker machine, will always go for a Royal if your original hand is a three card Royal, you could hit it after hitting zero, one, or two of the necessary cards on the first draw. Three parlay prices you could add up.

The more clever way to do this problem however, is to realize that you will hit it only if the next four cards contain the two you need. That could occur if the two cards were in position 1-2, 1-3, 2-3, or 3-4. Each has a probability of 2/47 × 1/46. Thus the answer is 8/2,162

Question: You laid $200 to $100 on a football game. Your team is now threatening to score and if they do, they will be 60 percent to win. If they don’t you make them 20%. You think they are 70% to score.  What is the fair price to get out of your bet at this moment.

Answer: You could get these answers via an algebra equation, but we show you a trick. Imagine that all the money is in a hat. In this case it is $300 of which you contributed $200. If you are 70% to be a 60% shot and 30% to be a 20% shot that makes you 48% to win. Which means you deserve 48% of that $300 hat. That’s $144.

But that means that you should lose $56 which is the fair amount to give the other guy if there is not really a hat. (You could also use the hat to answer the reverse question. If I settle by giving up $20 what am I implying are the true chances? Since I get $180 and you get $120 it’s supposed to mean I think I am a 3-2 favorite or 60%. ♠

David Sklansky is the author of The Theory of Poker, as well as nearly two dozen other guides on gambling, poker, and other games. The three-time WSOP bracelet winner’s latest book, Small Stakes No-Limit Hold’em: Help Them Give You Their Money, is now available on Amazon. You can contact Sklansky at [email protected].