A box contains 20 identical balls of which 10 are blue and 10 are green. The balls are drawn at random from the box one at a time with replacement. The probability that a blue ball is drawn 4th time on the 7th draw is

Options

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

Answer

To solve the problem of finding the probability that a blue ball is drawn for the fourth time on the seventh draw, we can follow these steps:

### Step 1: Understand the Problem
We have a box with 20 balls: 10 blue and 10 green. We are drawing balls with replacement, meaning the total number of balls remains constant. We want to find the probability that the fourth blue ball is drawn on the seventh draw.

### Step 2: Define the Probabilities
The probability of drawing a blue ball (P) is:
\[ P(\text{Blue}) = \frac{10}{20} = \frac{1}{2} \]
The probability of drawing a green ball (Q) is:
\[ P(\text{Green}) = \frac{10}{20} = \frac{1}{2} \]

### Step 3: Set Up the Conditions
For the fourth blue ball to be drawn on the seventh draw, we need:
- Exactly 3 blue balls to be drawn in the first 6 draws.
- The 7th draw must be a blue ball.

### Step 4: Use the Binomial Probability Formula
The number of ways to choose 3 successes (blue balls) in 6 trials (draws) can be calculated using the binomial coefficient:
\[ \binom{6}{3} \]
The probability of getting exactly 3 blue balls in 6 draws is given by:
\[ \left(\frac{1}{2}\right)^3 \left(\frac{1}{2}\right)^{3} = \left(\frac{1}{2}\right)^6 \]
Thus, the probability of drawing 3 blue balls in the first 6 draws is:
\[ \binom{6}{3} \left(\frac{1}{2}\right)^6 \]

### Step 5: Calculate the Binomial Coefficient
Calculate \(\binom{6}{3}\):
\[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]

### Step 6: Combine the Probabilities
Now, we combine the probabilities:
1. The probability of getting 3 blue balls in the first 6 draws:
\[ 20 \times \left(\frac{1}{2}\right)^6 \]
2. The probability of getting a blue ball on the 7th draw:
\[ \frac{1}{2} \]

### Step 7: Final Probability Calculation
The total probability that the fourth blue ball is drawn on the seventh draw is:
\[ P = 20 \times \left(\frac{1}{2}\right)^6 \times \left(\frac{1}{2}\right) = 20 \times \left(\frac{1}{2}\right)^7 = 20 \times \frac{1}{128} = \frac{20}{128} = \frac{5}{32} \]

### Final Answer
Thus, the probability that a blue ball is drawn for the fourth time on the seventh draw is:
\[ \frac{5}{32} \]