Two cards are drawn successively with replacement from a well shuffled deck of 52 cards. The probability distribution of the number of kings will be:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(P(0) = \displaystyle\frac{144}{169}\), \(P(1) = \displaystyle\frac{24}{169}\), \(P(2) = \displaystyle\frac{1}{169}\)
Let X be the number of kings. A deck has 4 kings, and the cards are drawn with replacement, so the two draws are independent with
\( \displaystyle P(\text{king}) = \frac{4}{52} = \frac{1}{13}, \quad P(\text{not king}) = \frac{12}{13} \)
X = 0 (no king in either draw):
\( \displaystyle P(X = 0) = \frac{12}{13} \times \frac{12}{13} = \frac{144}{169} \)
X = 1 (king first and not king second, or the other way round):
\( \displaystyle P(X = 1) = 2 \times \frac{1}{13} \times \frac{12}{13} = \frac{24}{169} \)
X = 2 (king in both draws):
\( \displaystyle P(X = 2) = \frac{1}{13} \times \frac{1}{13} = \frac{1}{169} \)
The three values add up to \(\frac{169}{169} = 1\), and they appear in this order only in Option (4).