Probabilities that A, B, C pass a test are `1/3, 1/4, 1/5` respectively. Probability that at least two of them will pass in a test is

Options

A. `1/60`
B. `1/6`
C. `47/60`
D. `1/5`

Answer

To find the probability that at least two of A, B, and C pass the test, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the probabilities of passing**:
- Probability that A passes: \( P(A) = \frac{1}{3} \)
- Probability that B passes: \( P(B) = \frac{1}{4} \)
- Probability that C passes: \( P(C) = \frac{1}{5} \)

2. **Calculate the probabilities of not passing**:
- Probability that A does not pass: \( P(A') = 1 - P(A) = 1 - \frac{1}{3} = \frac{2}{3} \)
- Probability that B does not pass: \( P(B') = 1 - P(B) = 1 - \frac{1}{4} = \frac{3}{4} \)
- Probability that C does not pass: \( P(C') = 1 - P(C) = 1 - \frac{1}{5} = \frac{4}{5} \)

3. **Calculate the probability of exactly two passing**:
- Case 1: A and B pass, C does not pass:
\[ P(A \cap B \cap C') = P(A) \cdot P(B) \cdot P(C') = \frac{1}{3} \cdot \frac{1}{4} \cdot \frac{4}{5} = \frac{1}{15} \]
- Case 2: A and C pass, B does not pass:
\[ P(A \cap B' \cap C) = P(A) \cdot P(B') \cdot P(C) = \frac{1}{3} \cdot \frac{3}{4} \cdot \frac{1}{5} = \frac{1}{20} \]
- Case 3: B and C pass, A does not pass:
\[ P(A' \cap B \cap C) = P(A') \cdot P(B) \cdot P(C) = \frac{2}{3} \cdot \frac{1}{4} \cdot \frac{1}{5} = \frac{1}{30} \]

4. **Calculate the probability of all three passing**:
\[ P(A \cap B \cap C) = P(A) \cdot P(B) \cdot P(C) = \frac{1}{3} \cdot \frac{1}{4} \cdot \frac{1}{5} = \frac{1}{60} \]

5. **Combine all probabilities**:
The total probability that at least two of them pass is the sum of the probabilities calculated in the previous steps:
\[ P(\text{at least 2 pass}) = P(A \cap B \cap C') + P(A \cap B' \cap C) + P(A' \cap B \cap C) + P(A \cap B \cap C) \]
\[ = \frac{1}{15} + \frac{1}{20} + \frac{1}{30} + \frac{1}{60} \]

6. **Finding a common denominator**:
The least common multiple of 15, 20, 30, and 60 is 60. Converting each fraction:
\[ \frac{1}{15} = \frac{4}{60}, \quad \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{30} = \frac{2}{60}, \quad \frac{1}{60} = \frac{1}{60} \]
Now, summing these:
\[ = \frac{4}{60} + \frac{3}{60} + \frac{2}{60} + \frac{1}{60} = \frac{10}{60} = \frac{1}{6} \]

### Final Answer:
The probability that at least two of A, B, and C pass the test is \( \frac{1}{6} \).