Question

Ruby picks 5 cards from a deck of 52 cards. We say that there are two distinct pairs (the hand “two pairs”) if the 5 cards contain 2 cards of a first denomination (i.e., one of “A”, “2”, ..., or “K”), 2 cards of a different denomination, and a single card of third different denomination. What is the probability of Ruby getting two distinct pairs? Assume all 5-card hands are equally likely

Answer 1

Two distinct pairs consists of 2 seperate pairs of 2 cards.

Step 1:

2 distinct demoninations can be chosen from 13 denomination in:

Step 2:

First Pair:

2 cards can be chosen from 4 cards in:

Step 3:

Scond Pair:

2 cards can be chosen from 4 cards in:

Step 4:

Fifth card- Denomination:

1 Demonimation can be chosen from remaining 13- 2 = 11 denominations in:

Step 5: Fifth card- Suits (Club, Spade, Heart, Diamond):

One suit can b chosen from 4 suits in:

Thus,

Total number of ways of getting 2 distinct pairs = 78 X 6 X 6 X 11 X 4 = 123552

Total number of choosing 5 cards :

So,

Probability of getting 2 distinct pairs = 123552/2598960 = 0.0475

So,

Answer is:

0.0475