Question

Four cards are drawn successively with replacement from well shuffled deck of 52 cards, then the probability that only two cards are club cards is ...........

• A. \(\frac{26}{128}\)
• B. \(\frac{24}{128}\)
• C. \(\frac{27}{128}\)
• D. \(\frac{28}{128}\)

Hint:

Each draw is a club with probability 1/4; use the binomial formula for exactly two successes in four trials.

Answer 1

The Correct Option is C

Step 1: Approach
Count the arrangements of two clubs (C) and two non-clubs (N) among four draws.

Step 2: Arrangements
There are 6 orders, such as CCNN, CNCN and so on. Each has the same probability $\dfrac14\cdot\dfrac14\cdot\dfrac34\cdot\dfrac34=\dfrac{9}{256}$.

Step 3: Total
$6\times\dfrac{9}{256}=\dfrac{27}{128}$. Option (C).

Final Answer:
The probability is 6 times (1/4)^2 times (3/4)^2, which is 27/128, option (C). \[ \boxed{\frac{27}{128}} \]