Find the probability of being dealt 5 cards from a standard 52-card deck, and getting one pair.

Question

I know there are 2598960 ways to draw 5 cards.

I tried a few different ways to find the complete answer but I got it wrong every time.

Answer 1

A typical hand would be 2 of one value and 3 other cards of all different values

 

To get 2 of one value you could do 13*₄C₂ then you would multiply this by ₁₂C₃*₄C₁*₄C_1*4C_{1  .}

 

The first term above, as Ira said, represents 13 ways of choosing a pair of cards of the same value.

 

However the right expression represents the number of combinations (order does not matter) of 3 out of the 12 remaining values and then selecting 1 of each of those 3 values chosen.

 

If you need more explanation please comment back.

Answer 2

There are 13 different types of pairs.....2 kings, 2 queens.....

 

So see if this makes sense to you.

₁₃C₁ * ₄C₂     *     ₁₂C₁ * ₄C₁     *      ₁₁C₁*₄C₁     *    ₁₀C₁*₄C₁       *      ₉C₁*₄C₁

 

Hope this helped