Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. The probability distribution of number of aces is given by:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(P(0) = \frac{144}{169},\ P(1) = \frac{24}{169},\ P(2) = \frac{1}{169}\)
Let X be the number of aces. The cards are drawn with replacement, so the two draws are independent, and in each draw
\(\displaystyle P(\text{ace}) = \frac{4}{52} = \frac{1}{13},\quad P(\text{not an ace}) = \frac{12}{13}\)
No ace in either draw:
\(\displaystyle P(X = 0) = \frac{12}{13} \times \frac{12}{13} = \frac{144}{169}\)
Exactly one ace: the ace comes either in the first draw only or in the second draw only, so there are two cases:
\(\displaystyle P(X = 1) = 2 \times \frac{1}{13} \times \frac{12}{13} = \frac{24}{169}\)
Two aces:
\(\displaystyle P(X = 2) = \frac{1}{13} \times \frac{1}{13} = \frac{1}{169}\)
As a check, \(144 + 24 + 1 = 169\), so the probabilities add up to 1.