Card Game Probabilities
Date: 5/25/96 at 17:50:59 From: Jim Funk Subject: Poker My question is this: what is the probability of getting two of a kind in a five-card Poker hand? Jim Funk
Date: 9/1/96 at 18:27:47 From: Doctor James Subject: Re: Poker Let's say you are dealt 5 cards, one at a time, and these are the first cards dealt from the deck. The first card you are dealt has a 100 percent chance of not duplicating any other card (since there are no other cards out!). Now there are 3 cards out of 51 remaining in the deck of that rank. So the next card has a 48/51 chance of not giving you two of a kind. Assuming that's the case, there are now 6 cards out of 50 that will give you two of a kind (three from the first, and three from the second). So the next card has a 44/50 chance of not giving you a two of a kind. The fourth card has a 9/49 chance of duplicating another card, or a 40/49 chance of not giving you two of a kind. The fifth card has a 36/48 chance of not giving you two of a kind. Multiplying them all together gives you a 2112/4165, or 50.7 percent chance of getting no duplicates of any kind in your hand. Thus, you have a 49.3 percent chance of getting some kind of duplicate (2 of a kind, 3 or 4 of a kind). You can calculate the probability of getting a three of four of a kind, and subtract that probability from the 49.3 percent to find the probability of getting a two of a kind, but that gets much more complicated. -Doctor James, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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