A box contains 20 identical balls of which 10 balls are red . The balls are drawn at random from the box oone at a time with replacement .The probability that a white ball is drawn for the 4th time on the 7th draw is

Options

A. `5/64`
B. `5/32`
C. `27/32`
D. `1/2`

Answer

To find the probability that a white ball is drawn for the 4th time on the 7th draw, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the total number of balls and their colors**:
- The box contains 20 balls in total: 10 red and 10 white.

2. **Calculate the probability of drawing a white ball (W) and a red ball (R)**:
- Probability of drawing a white ball, \( P(W) = \frac{10}{20} = \frac{1}{2} \)
- Probability of drawing a red ball, \( P(R) = \frac{10}{20} = \frac{1}{2} \)

3. **Understand the scenario**:
- We want the 4th white ball to be drawn on the 7th draw. This means that in the first 6 draws, we must have exactly 3 white balls drawn.

4. **Use the binomial probability formula**:
- We need to calculate the probability of getting exactly 3 white balls in the first 6 draws. This can be modeled using the binomial distribution:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]
where:
- \( n = 6 \) (the number of draws),
- \( k = 3 \) (the number of successful draws, i.e., white balls),
- \( p = \frac{1}{2} \) (the probability of drawing a white ball).

5. **Calculate the binomial coefficient**:
- Calculate \( \binom{6}{3} \):
\[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]

6. **Calculate the probabilities**:
- The probability of drawing 3 white balls and 3 red balls in the first 6 draws:
\[ P(\text{3 W in 6 draws}) = \binom{6}{3} \left(\frac{1}{2}\right)^3 \left(\frac{1}{2}\right)^{3} = 20 \cdot \left(\frac{1}{2}\right)^6 = 20 \cdot \frac{1}{64} = \frac{20}{64} = \frac{5}{16} \]

7. **Account for the 7th draw**:
- The probability that the 7th draw is a white ball (the 4th white ball) is \( \frac{1}{2} \).

8. **Combine the probabilities**:
- The total probability that the 4th white ball is drawn on the 7th draw:
\[ P(\text{4th W on 7th draw}) = P(\text{3 W in 6 draws}) \times P(\text{W on 7th draw}) = \frac{5}{16} \times \frac{1}{2} = \frac{5}{32} \]

### Final Answer:
The probability that a white ball is drawn for the 4th time on the 7th draw is \( \frac{5}{32} \).