### Probability Games

A colleague of the mathematics brand posed this question to me today: Say you're playing five card draw poker and you have two tens. You're playing with four other players and you get the feeling you don't have the strongest hand. What are the odds that, if you discard three, you'll draw another ten?

Answers? Anyone? Bueller?

We both agreed that in simple terms you'd say (before you drew the first card), you have a 2 in 47 chance (there's two tens left and there's 47 cards left which you haven't seen). If after the first card drawn you didn't have a ten, you would have a 2 in 46 chance and then a 2 in 45 chance for the third.

But he argued that because other players already have cards that the probability changes--that you have LESS cards out there to draw from. I guess if you were God and could see everybody's cards, you could then calculate the odds. But he was saying, realistically, the probability is different even though you don't KNOW what the other cards are.

I'm not sure what I think about that. I mean, if you have that hand, it doesn't matter what cards are in other peoples hands or in the deck--there are 47 cards you haven't seen. But somehow, the idea of odds changing simply by the redistribution of the cards is intriguing--even if it is a crock of crapola.

If you sensed that the other players had a better hand than a pair of tens, you'd have to figure there are jacks, aces, queens, etc. out there and you could take that into consideration. But that's not what my poker buddy was talking about. I guess it's a philosophical question...if a tree farts in the woods and nobody's there to hear it, does it happen? If there are other cards in player's hands which you can't see does probability change? Philosophy and mathematics...it's like chicken and waffles...tasty but kind of messed up.

## 1 Comments:

Hi Jamie. My mom says:

As the daughter, and widow, of poker players; and, having learned my numbers by sitting in the lap's of my Dad's friends, as a child, while they were playing, I would like to address your friend's question. Given his hypothesis, it would/could mean that you would discount the 20 cards that are being held by the other players. Which would mean that your odds would then be 27 to 2, 26 to 2, 25 to 2, giving you a much better chance of pulling another ten then 47 to 2, etc. Since we know, through years of playing, that statistically 4 of a kind, or even 3 of a kind, is a rare hand with a 5 player game, even philosphically it is hard to accept the concept that we are better off believing that we have a better chance of pulling another ten because more people are holding cards that we can't see. I think an even better example of that premise is what we thought the odds were of Bush being reelected, based on a lot of new voters. Anyway, I enjoyed the problem. Thanks a lot.

Post a Comment

<< Home