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

Options

A. `(9)/(64)`
B. `(1)/(64)`
C. `(1)/(16)`
D. `(9)/(16)`

Answer

To solve the problem, we need to determine the probability that a person fails to draw a spade in the first two attempts when drawing cards from a standard deck of playing cards.

### Step-by-Step Solution:

1. **Understanding the Deck**:
A standard deck of playing cards contains 52 cards. Out of these, there are 13 spades.

2. **Calculating the Probability of Drawing a Spade**:
The probability of drawing a spade in one attempt is given by the formula:
\[ P(\text{Spade}) = \frac{\text{Number of Spades}}{\text{Total Number of Cards}} = \frac{13}{52} = \frac{1}{4} \]

3. **Calculating the Probability of Failing to Draw a Spade**:
The probability of not drawing a spade (failure) in one attempt is:
\[ P(\text{Not Spade}) = 1 - P(\text{Spade}) = 1 - \frac{1}{4} = \frac{3}{4} \]

4. **Calculating the Probability of Failing Twice**:
Since the draws are independent (the card is replaced each time), the probability of failing to draw a spade in the first two attempts is:
\[ P(\text{Fail first two times}) = P(\text{Not Spade}) \times P(\text{Not Spade}) = \left(\frac{3}{4}\right) \times \left(\frac{3}{4}\right) = \left(\frac{3}{4}\right)^2 = \frac{9}{16} \]

### Final Answer:
The probability that he will fail to draw a spade in the first two attempts is \(\frac{9}{16}\).