The probabilities of A, B, C solving a question are `(1)/(3),(2)/(7)` and `(3)/(8)` respectively. Find the probability that exactly one of them will solve it.

Answer

To find the probability that exactly one of A, B, or C will solve the question, we will follow these steps:

### Step 1: Define the probabilities
Let:
- \( P(A) = \frac{1}{3} \)
- \( P(B) = \frac{2}{7} \)
- \( P(C) = \frac{3}{8} \)

### Step 2: Calculate the probabilities of not solving
We need to find the probabilities that each of them does not solve the question:
- \( P(A') = 1 - P(A) = 1 - \frac{1}{3} = \frac{2}{3} \)
- \( P(B') = 1 - P(B) = 1 - \frac{2}{7} = \frac{5}{7} \)
- \( P(C') = 1 - P(C) = 1 - \frac{3}{8} = \frac{5}{8} \)

### Step 3: Calculate the probability of exactly one solving
The probability that exactly one of them solves the question can be calculated by considering three scenarios:
1. A solves it, while B and C do not.
2. B solves it, while A and C do not.
3. C solves it, while A and B do not.

The probabilities for these scenarios are:
1. \( P(A) \cdot P(B') \cdot P(C') = \frac{1}{3} \cdot \frac{5}{7} \cdot \frac{5}{8} \)
2. \( P(A') \cdot P(B) \cdot P(C') = \frac{2}{3} \cdot \frac{2}{7} \cdot \frac{5}{8} \)
3. \( P(A') \cdot P(B') \cdot P(C) = \frac{2}{3} \cdot \frac{5}{7} \cdot \frac{3}{8} \)

### Step 4: Calculate each probability
Now we calculate each of these probabilities:
1. \( P(A) \cdot P(B') \cdot P(C') = \frac{1}{3} \cdot \frac{5}{7} \cdot \frac{5}{8} = \frac{25}{168} \)
2. \( P(A') \cdot P(B) \cdot P(C') = \frac{2}{3} \cdot \frac{2}{7} \cdot \frac{5}{8} = \frac{20}{168} \)
3. \( P(A') \cdot P(B') \cdot P(C) = \frac{2}{3} \cdot \frac{5}{7} \cdot \frac{3}{8} = \frac{30}{168} \)

### Step 5: Sum the probabilities
Now we sum these probabilities to find the total probability that exactly one of them solves the question:
\[ P(\text{exactly one solves}) = \frac{25}{168} + \frac{20}{168} + \frac{30}{168} = \frac{75}{168} \]

### Step 6: Simplify the fraction
Now, we simplify \( \frac{75}{168} \):
- The GCD of 75 and 168 is 3.
- Thus, \( \frac{75 \div 3}{168 \div 3} = \frac{25}{56} \).

### Final Answer
The probability that exactly one of A, B, or C will solve the question is \( \frac{25}{56} \).

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