A cricket team has 15 members, of whom only 5 can bowl. If the names of the 15 members are put into a hat and 11 drawn random, then the chance of obtaining an eleven containig at least 3 bowlers

Options

A. `(7)/(13)`
B. `(11)/(15)`
C. `(12)/(13)`
D. none

Answer

To find the probability of obtaining an eleven containing at least 3 bowlers from a cricket team of 15 members (5 of whom can bowl), we can break down the problem into manageable steps.

### Step-by-Step Solution:

1. **Identify Total Members and Bowlers**:
- Total members = 15
- Bowlers = 5
- Non-bowlers (batsmen) = 15 - 5 = 10

2. **Define the Event**:
- We want to find the probability of selecting 11 members that contain at least 3 bowlers.

3. **Possible Cases**:
- The cases for selecting at least 3 bowlers are:
- Case 1: Exactly 3 bowlers and 8 batsmen
- Case 2: Exactly 4 bowlers and 7 batsmen
- Case 3: Exactly 5 bowlers and 6 batsmen

4. **Calculate Total Ways to Choose 11 Members**:
- The total ways to choose 11 members from 15 is given by the combination formula:
\[ \text{Total ways} = \binom{15}{11} \]

5. **Calculate Each Case**:
- **Case 1**: Exactly 3 bowlers and 8 batsmen
\[ \text{Ways} = \binom{5}{3} \times \binom{10}{8} \]
- **Case 2**: Exactly 4 bowlers and 7 batsmen
\[ \text{Ways} = \binom{5}{4} \times \binom{10}{7} \]
- **Case 3**: Exactly 5 bowlers and 6 batsmen
\[ \text{Ways} = \binom{5}{5} \times \binom{10}{6} \]

6. **Calculate the Combinations**:
- Calculate each combination:
- \(\binom{5}{3} = 10\), \(\binom{10}{8} = 45\) → Case 1: \(10 \times 45 = 450\)
- \(\binom{5}{4} = 5\), \(\binom{10}{7} = 120\) → Case 2: \(5 \times 120 = 600\)
- \(\binom{5}{5} = 1\), \(\binom{10}{6} = 210\) → Case 3: \(1 \times 210 = 210\)

7. **Total Ways for At Least 3 Bowlers**:
- Total for at least 3 bowlers = Case 1 + Case 2 + Case 3
\[ \text{Total ways for at least 3 bowlers} = 450 + 600 + 210 = 1260 \]

8. **Calculate Total Combinations**:
- Total ways to choose 11 from 15:
\[ \binom{15}{11} = \binom{15}{4} = 1365 \]

9. **Calculate Probability**:
- The probability of selecting at least 3 bowlers:
\[ P(\text{at least 3 bowlers}) = \frac{\text{Total ways for at least 3 bowlers}}{\text{Total ways to choose 11}} = \frac{1260}{1365} \]

10. **Simplify the Probability**:
- Simplifying \(\frac{1260}{1365}\):
\[ P(\text{at least 3 bowlers}) = \frac{12}{13} \]

### Final Answer:
The probability of obtaining an eleven containing at least 3 bowlers is \(\frac{12}{13}\).