A hand of 8 cards is chosen at random from an ordinary deck

A hand of 8 cards is chosen at random from an ordinary deck of 52 playing cards without replacement.

a) what is the probability that the hand does not have any hearts?

b) What is the probability that the hand consists of two hearts, two diamonds, two clubs, and two spades?

Solution

a. In choosing 8 cards out of 52, the number of ways you choose the cards such that the chosen 8 cards do not include an y hearts = No. of ways of choosing 8 cards from the 39 non-heart cards = 39C8

Total no. of ways = 52C8

Probability that the hand does not have any hearts = 39C8 / 52C8 = 0.0818

b. In choosing 8 cards out of 52, the number of ways you choose the cards such that the chosen 8 cards have two hearts, two diamonds, two clubs, and two spades

= No. of ways of choosing 2 cards from the 13 hearts * No. of ways of choosing 2 cards from the 13 diamonds * No. of ways of choosing 2 cards from the 13 clubs * No. of ways of choosing 2 cards from the 13 spades

= 13C2 * 13C2 * 13C2 * 13C2

Probability = 13C2 * 13C2 * 13C2 * 13C2 / 52C8 = 0.0492