Linda Mae strode into our local poker room, signed up for a few games, sat next to me and asked why, when she had a suited hand at hold 'em, the probability that the dealer would lay a trump next on the turn was 9/47.
Stunned that she would ask about probabilities, (you may recall that she's hopeless at mathematics) I sat there like a lump until I recovered enough to ask what troubled her.
"Well," she said, "you know there must be some hearts in the hands of my opponents. So there can't be 9 hearts left in the deck. So the probability can't be 9/47."
It took me a while to ascertain the source of her difficulty. Finally, I said, "You're right, there probably are some hearts dealt among your opponents. In a nine-handed game the probability of hearts among the 18 face-down cards, 2 to each opponent and 2 burn cards, equals 18*9/47, or 3.4468 hearts, on the average.
Triumphantly, she exclaimed, "I knew it! I knew all those experts, even you, were full of it, because the probability of another heart on the turn isn't really 9/47."
"Hold on," I cautioned her, "suppose you held 10h-9h and the dealer flopped Ah-4c-2h. We know that 3.45 hearts, on the average, lie among your opponents' cards and the two burn cards. Therefore, on the average, there are about 5.55 hearts left in the deck. How many cards does the dealer have left," I asked.
"The nine of us hold 18 cards. There are 3 cards on the flop and 2 burn cards, so she has 29 cards."
"Right. And if there are 5.55 hearts in the 29 remaining cards, then what's the probability that the turn card will be a heart," I asked.
Linda Mae thought for a bit and answered, "It's 5.55/29. That's easy enough."
"Right," I told her, "now what's that equal in decimals?"
She rummaged around in her enormous Gucci™ bag, fished out a calculator, and punched in the numbers. "It's 0.1914," she answered. I then asked her to calculate the probability, 9/47, in decimals. "That's 0.1915," she said, surprised. "That's about the same as 5.55 divided by 29!"
"If you did the calculations with more decimal places," I told her, "then you'd find they're exactly equal.
"To your way of looking at it, because there are 3+ hearts dealt among your opponents, the numerator changes," I observed, "but you forgot that the denominator changes too. In fact, no matter how you slice it, both numerator and denominator change such that the probability is always 0.191489, or 9/47."
Somewhat deflated, she still stuck with her point of view and asked, "Why do you mathematicians say there are 47 unknown cards when they really aren't? The other 8 players know the whereabouts of 16 of them."
"If you could peek at their cards, or if the table were equipped with miniature cameras and you were in the viewing audience, then you would know all their cards, and you could compute the exact probability of another heart on the turn," I answered. "It would be different from 9/47. But there's no peeking allowed, so there are 47 unknown cards."
Linda Mae pondered for a while and then exclaimed brightly, "When you say there are 47 unknown cards, you really mean there are 47 cards unknown to you! And since you don't know where the other 9 hearts are either, that's why you use 9/47 for the probability!" I agreed. "Then why didn't you say so in the first place?!"
"Because, I answered, "like Dorothy in the Land of Oz, you had to find that out for yourself."
Mr. Burke is the author of Flop: The Art of Winning at Low-Limit Hold 'Em, on sale at amazon & kokopellipress.com. E-mail your Hold 'Em questions to richardburke@comcast.net
