Question

A 5-card hand is dealt from a well-shuffled deck of playing cards.

What is the probability of getting a hand with three cards of the same rank?

What is the probability of getting a hand with two cards of the same rank?

Please write as legibly as possible

Answer 1

First, count the number of five-card hands that can be dealt from a standard deck of 52 cards.

Total number of distinct poker hands= 52C5 = 2,598,960

Let's understand the ranks in the deck of cards.

There are 13 ranks. In each rank there are 4 cards(one in each suit). There are 52 cards in the pack, and the ranking of the individual cards, from high to low, is ace, king, queen, jack, 10, 9, 8, 7, 6, 5, 4, 3, 2. (13 ranks).

A)there are 13 different ranks with each having 4 cards.

So, number of ways to get 3 cards of same rank in a 5 card hand = number of ways to get 3 cards of same rank and other two cards 9f different ranks = 13×(12×11) = 1716

(since, first 3 cards of same rank can be any one of the 13 different rank cards and the remaining two can be any rank card except those belonging to the chosen rank for first 3 cards i.e. two out of remaining 12 rank cards

So, 4th card chosen in 12 ways and 5th card chosen in 11 ways)

So, probability of getting 3 cards of same rank = 1716/2598960 = 0.0006602641

B) number of ways to get 2 cards of same rank = Number of ways to get 2 cards of same rank and remaining 3 cards are of different rank =13 ×(12×11× 10 )= 17160

Where first 2 cards of same rank can be chosen in 13 ways (any one of the 13 different rank card)

, the 3rd card can be chosen in 12 ways as 12 different rank cards are present now.

,similarly, the 4th cards can be chosen in 11 ways and 5th card can be chosen in 10 ways

So, probability of getting 2 cards of same rank = 17160/2598960 = 0.006602641