Question

Three cards are drawn successively without replacement from a pack of 52 well shuffled cards. What is the probability that first two cards are kings and the third card drawn is an ace?

• A. \(\frac{4}{52}\)
• B. \(\frac{3}{51}\)
• C. \(\frac{4}{50}\)
• D. \(\frac{2}{5525}\)

Hint:

For draws without replacement, update both the numerator and denominator after every draw because the sample space changes.

Answer 1

The Correct Option is D

Concept: When cards are drawn without replacement, the events are dependent. Therefore, \[ P(A\cap B\cap C) = P(A)\times P(B|A)\times P(C|A\cap B). \] A standard deck contains \[ 52 \] cards, including \[ 4 \] kings and \[ 4 \] aces.

Step 1: Find the probability that the first card is a king. \[ P(K_1)=\frac{4}{52}. \]

Step 2: Find the probability that the second card is also a king. After one king is removed, \[ P(K_2|K_1) = \frac{3}{51}. \]

Step 3: Find the probability that the third card is an ace. After two kings have been removed, there are still four aces among fifty cards. \[ P(A_3|K_1\cap K_2) = \frac{4}{50}. \]

Step 4: Multiply the probabilities. \[ P = \frac{4}{52} \times \frac{3}{51} \times \frac{4}{50}. \] \[ = \frac{1}{13} \times \frac{1}{17} \times \frac{2}{25}. \] \[ = \frac{2}{13\times17\times25}. \] \[ = \frac{2}{5525}. \] Conclusion: \[ \boxed{\frac{2}{5525}} \]