Answer to Question #261051 in Discrete Mathematics for Alina

Question

Answer the following:

(a) You have a standard 52 card deck. If you draw two cards at random, what is the probability that they are both hearts?

(b) You reshuffle your cards into the deck, but unfortunately now a mischievous dog (a golden retriever named Bubbles), decides to eat one of the cards! Assuming that the missing card is a diamond, now when you draw two cards at random what is the probability that they are both hearts?

Answer 1

(a)

There are 13 hearts in a full deck of 52 cards.

$$P(2\ hearts)=\dfrac{\dbinom{13}{2}}{\dbinom{52}{2}}=\dfrac{\dfrac{13!}{2!(13-2)!}}{\dfrac{52!}{2!(52-50)!}}$$

$$=\dfrac{13(12)}{52(51)}=\dfrac{1}{17}$$

(b)

$$P(2\ hearts)=\dfrac{\dbinom{13}{2}}{\dbinom{52-1}{2}}=\dfrac{\dfrac{13!}{2!(13-2)!}}{\dfrac{51!}{2!(51-50)!}}$$

$$=\dfrac{13(12)}{51(50)}=\dfrac{26}{425}$$