There are 3 people A,B,C probability that A speaks truth is 3/10 B speaks is 3/7 and C speaks 5/6 for particular kind of question at most 2 people speak truth .What is the probability that B will speak truth for particular question asked?

Options

A. (8/19)
B. (9/28)
C. (5/23)
D. (11/31)

Answer

To solve the problem, we need to find the probability that person B speaks the truth given that at most two people (A, B, and C) can speak the truth.

### Step-by-Step Solution:

1. **Identify the Probabilities**:
- Probability that A speaks the truth, \( P(A) = \frac{3}{10} \)
- Probability that B speaks the truth, \( P(B) = \frac{3}{7} \)
- Probability that C speaks the truth, \( P(C) = \frac{5}{6} \)

2. **Calculate the Probabilities of Speaking False**:
- Probability that A speaks false, \( P(A') = 1 - P(A) = 1 - \frac{3}{10} = \frac{7}{10} \)
- Probability that B speaks false, \( P(B') = 1 - P(B) = 1 - \frac{3}{7} = \frac{4}{7} \)
- Probability that C speaks false, \( P(C') = 1 - P(C) = 1 - \frac{5}{6} = \frac{1}{6} \)

3. **Define the Cases**:
- **Case 1**: Only B speaks the truth (A and C speak false).
- **Case 2**: A and B speak the truth (C speaks false).
- **Case 3**: B and C speak the truth (A speaks false).

4. **Calculate the Probability for Each Case**:
- **Case 1**:
\[ P(B \text{ speaks truth}) = P(B) \times P(A') \times P(C') = \frac{3}{7} \times \frac{7}{10} \times \frac{1}{6} = \frac{3 \times 7 \times 1}{7 \times 10 \times 6} = \frac{3}{60} = \frac{1}{20} \]

- **Case 2**:
\[ P(A \text{ and } B \text{ speak truth}) = P(A) \times P(B) \times P(C') = \frac{3}{10} \times \frac{3}{7} \times \frac{1}{6} = \frac{3 \times 3 \times 1}{10 \times 7 \times 6} = \frac{9}{420} = \frac{3}{140} \]

- **Case 3**:
\[ P(B \text{ and } C \text{ speak truth}) = P(B) \times P(C) \times P(A') = \frac{3}{7} \times \frac{5}{6} \times \frac{7}{10} = \frac{3 \times 5 \times 7}{7 \times 6 \times 10} = \frac{15}{60} = \frac{1}{4} \]

5. **Combine the Probabilities**:
\[ P(B \text{ speaks truth}) = P(\text{Case 1}) + P(\text{Case 2}) + P(\text{Case 3}) = \frac{1}{20} + \frac{3}{140} + \frac{1}{4} \]

6. **Find a Common Denominator**:
The least common multiple of 20, 140, and 4 is 140.
- Convert each fraction:
- \( \frac{1}{20} = \frac{7}{140} \)
- \( \frac{3}{140} = \frac{3}{140} \)
- \( \frac{1}{4} = \frac{35}{140} \)

7. **Add the Fractions**:
\[ P(B \text{ speaks truth}) = \frac{7}{140} + \frac{3}{140} + \frac{35}{140} = \frac{45}{140} \]

8. **Simplify the Fraction**:
\[ \frac{45}{140} = \frac{9}{28} \]

### Final Answer:
The probability that B will speak the truth is \( \frac{9}{28} \).