Question

If two cards are drawn at random from an ordinary deck of 52 playing cards, what are the probabilites that they will both be hearts if the first card is replaced before the second card is drawn? and what is the probability if the first card is not replaced before the second card is drawn?

Answer 1

With replacement: P[first card a heart] = 0.25 and P[second card a heart] = 0.25
So P[both hearts] = 0.25*0.25 = 1/16

Without replacement: P[first card a heart] = 0.25 and P[second card a heart] = 12/52
So P[both hearts] = 0.25*(12/52)= 3/28

This follows because if we are successful on our first draw, there will be only 12 hearts remaining out of 52 cards for the second draw.