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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
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Associated Topics:
High School Probability

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