From a group of 5 boys and 3 girls, three persons are chosen at random. Find the probability that there are more girls than boys

Options

A. `(3)/(8)`
B. `(4)/(7)`
C. `(5)/(8)`
D. `(2)/(7)`

Answer

To find the probability that there are more girls than boys when choosing 3 persons from a group of 5 boys and 3 girls, we can follow these steps:

### Step 1: Determine the total number of ways to choose 3 persons from the group.
The total number of persons in the group is:
- Boys: 5
- Girls: 3
- Total = 5 + 3 = 8 persons

The number of ways to choose 3 persons from 8 is given by the combination formula:
\[ \text{Total ways} = \binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \]

### Step 2: Identify the favorable cases where girls are more than boys.
To have more girls than boys in a selection of 3 persons, the possible combinations are:
1. 3 girls and 0 boys
2. 2 girls and 1 boy

### Step 3: Calculate the number of ways for each favorable case.
1. **Case 1: 3 girls and 0 boys**
- We can choose all 3 girls from the 3 available:
\[ \text{Ways} = \binom{3}{3} = 1 \]

2. **Case 2: 2 girls and 1 boy**
- We can choose 2 girls from 3 and 1 boy from 5:
\[ \text{Ways} = \binom{3}{2} \times \binom{5}{1} = 3 \times 5 = 15 \]

### Step 4: Calculate the total number of favorable cases.
Adding the ways from both cases:
\[ \text{Total favorable cases} = 1 + 15 = 16 \]

### Step 5: Calculate the probability.
The probability \( P \) that there are more girls than boys is given by the ratio of favorable cases to total cases:
\[ P = \frac{\text{Favorable cases}}{\text{Total cases}} = \frac{16}{56} \]

### Step 6: Simplify the probability.
To simplify \( \frac{16}{56} \):
\[ P = \frac{16 \div 8}{56 \div 8} = \frac{2}{7} \]

### Final Answer:
The probability that there are more girls than boys when choosing 3 persons from the group is \( \frac{2}{7} \).

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