Probability Mass Function

Question

Cards are selected at random and with replacement from an ordinary deck of 52 cards. Define the random variable X as the number of trials until we obtain the first ace. Give the probability mass function of X and compute the probability that we need four trials to get the first ace.

Answer 1

Selection with replacement means you're drawing from the whole deck every time. Four aces in the deck means Prob(Ace) = 4/52 = 1/13 for any given draw. Prob(not Ace) = 48/52 = 12/13.

Using independence, Prob(1st ace on nth draw) = Prob(non-Ace on all draws prior to nth) x Prob(Ace on nth draw) = (12/13)^(n-1) x (1/13). This is the probability mass function of X.

Prob(1st ace on 4th draw) = (12/13)³ x (1/13)