A committee of 4 persons has to be chosen from 8 boys and 6 girls, consisting of at least one girl. Find the probability that the committee consists of more girls than boys.

Answer

To solve the problem of finding the probability that a committee of 4 persons chosen from 8 boys and 6 girls consists of more girls than boys, we can follow these steps:

### Step 1: Identify the possible combinations of girls and boys
To have more girls than boys in a committee of 4, the possible combinations are:
1. 4 girls and 0 boys
2. 3 girls and 1 boy

### Step 2: Calculate the number of ways to choose 4 girls and 0 boys
The number of ways to choose 4 girls from 6 is given by the combination formula \( \binom{n}{r} \), which is calculated as:
\[ \binom{6}{4} = \frac{6!}{4!(6-4)!} = \frac{6 \times 5}{2 \times 1} = 15 \]
The number of ways to choose 0 boys from 8 is:
\[ \binom{8}{0} = 1 \]
Thus, the total ways to choose 4 girls and 0 boys is:
\[ \text{Total ways} = \binom{6}{4} \times \binom{8}{0} = 15 \times 1 = 15 \]

### Step 3: Calculate the number of ways to choose 3 girls and 1 boy
The number of ways to choose 3 girls from 6 is:
\[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]
The number of ways to choose 1 boy from 8 is:
\[ \binom{8}{1} = 8 \]
Thus, the total ways to choose 3 girls and 1 boy is:
\[ \text{Total ways} = \binom{6}{3} \times \binom{8}{1} = 20 \times 8 = 160 \]

### Step 4: Calculate the total number of favorable outcomes
The total number of favorable outcomes (committees with more girls than boys) is:
\[ \text{Favorable outcomes} = 15 + 160 = 175 \]

### Step 5: Calculate the total number of ways to choose any 4 persons from 14
The total number of ways to choose any 4 persons from 14 (8 boys + 6 girls) is:
\[ \binom{14}{4} = \frac{14!}{4!(14-4)!} = \frac{14 \times 13 \times 12 \times 11}{4 \times 3 \times 2 \times 1} = 1001 \]

### Step 6: Calculate the probability
The probability that the committee consists of more girls than boys is given by the ratio of favorable outcomes to the total outcomes:
\[ P(\text{more girls than boys}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{175}{1001} \]

### Final Answer
Thus, the probability that the committee consists of more girls than boys is:
\[ \frac{175}{1001} \]