A committee of 4 boys and 3 girls is chosen from 8 boys and 5 girls. One of the boys is a brother of one of the girls. Probability that both are in the committee is

Options

A. `0.3`
B. `0.4`
C. `0.5`
D. `0.8`

Answer

To solve the problem of finding the probability that both a brother and sister are in a committee of 4 boys and 3 girls chosen from 8 boys and 5 girls, we can follow these steps:

### Step 1: Identify the total number of boys and girls
We have:
- Total boys = 8
- Total girls = 5

### Step 2: Determine the committee composition
We need to form a committee of:
- 4 boys
- 3 girls

### Step 3: Calculate the total ways to form the committee
The total number of ways to choose 4 boys from 8 and 3 girls from 5 can be calculated using combinations:

\[ \text{Total ways} = \binom{8}{4} \times \binom{5}{3} \]

Calculating these values:
- \(\binom{8}{4} = \frac{8!}{4! \cdot 4!} = 70\)
- \(\binom{5}{3} = \frac{5!}{3! \cdot 2!} = 10\)

Thus, the total ways to form the committee is:
\[ \text{Total ways} = 70 \times 10 = 700 \]

### Step 4: Calculate the favorable outcomes
Now, we need to find the number of ways to form the committee such that both the brother (let's call him B) and the sister (let's call her G) are included.

1. **Select B and G**: Since B and G are already chosen, we need to select:
- 3 more boys from the remaining 7 boys (since one boy is already chosen)
- 2 more girls from the remaining 4 girls (since one girl is already chosen)

The number of ways to select the remaining boys and girls is:
\[ \text{Favorable ways} = \binom{7}{3} \times \binom{4}{2} \]

Calculating these values:
- \(\binom{7}{3} = \frac{7!}{3! \cdot 4!} = 35\)
- \(\binom{4}{2} = \frac{4!}{2! \cdot 2!} = 6\)

Thus, the favorable ways to form the committee is:
\[ \text{Favorable ways} = 35 \times 6 = 210 \]

### Step 5: Calculate the probability
The probability that both the brother and sister are in the committee is given by the ratio of favorable outcomes to total outcomes:

\[ P(\text{B and G in committee}) = \frac{\text{Favorable ways}}{\text{Total ways}} = \frac{210}{700} = \frac{3}{10} \]

### Final Answer
Thus, the probability that both the brother and sister are in the committee is:

\[ \frac{3}{10} \text{ or } 0.3 \]

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