Question

A person draws a card from a pack of playing cards, replaces it, and shuffles the pack. He continues doing this until he draws a spade. The chance that he will fail exactly the first two times is

• A. $9/64$
• B. $1/64$
• C. $1/16$
• D. $9/16$

Answer 1

The correct option is D

$9/16$

Explanation for the correct option:

Calculate the probability that he fails exactly the first two times :

There are 52 cards in a pack.

The probability of getting a spade $=\frac{{}^{13}C_1}{{}^{52}C_1}=\frac{13}{52}=\frac{1}{4}$

then, the probability of not getting a spade $=1-\frac{1}{4}=\frac{3}{4}$

Since he fails exactly the first two times.

Therefore, the probability that he fails exactly first two times $=\frac{3}{4}\times \frac{3}{4}=\frac{9}{16}$

Hence, the correct option is (D).