`A` speaks truth in `75%` cases and `B` speaks truth in `80%` cases. Probability that they contradict each other in statement, is

Options

A. a) `7/(20)`
B. b) `(13)/(20)`
C. c) `3/5`
D. d) `2/5`

Explanation

To find the probability that A and B contradict each other in their statements, we can follow these steps:

### Step 1: Determine the probabilities of A and B speaking the truth and lying.
- A speaks the truth in 75% of cases, so:
\[ P(A \text{ speaks truth}) = \frac{3}{4} \]
Therefore, the probability that A lies is:
\[ P(A \text{ lies}) = 1 - P(A \text{ speaks truth}) = 1 - \frac{3}{4} = \frac{1}{4} \]

- B speaks the truth in 80% of cases, so:
\[ P(B \text{ speaks truth}) = \frac{4}{5} \]
Therefore, the probability that B lies is:
\[ P(B \text{ lies}) = 1 - P(B \text{ speaks truth}) = 1 - \frac{4}{5} = \frac{1}{5} \]

### Step 2: Identify the scenarios where A and B contradict each other.
There are two scenarios where A and B contradict each other:
1. A speaks the truth while B lies.
2. A lies while B speaks the truth.

### Step 3: Calculate the probabilities for each scenario.
- For the first scenario (A speaks truth and B lies):
\[ P(A \text{ speaks truth}) \times P(B \text{ lies}) = \frac{3}{4} \times \frac{1}{5} = \frac{3}{20} \]

- For the second scenario (A lies and B speaks truth):
\[ P(A \text{ lies}) \times P(B \text{ speaks truth}) = \frac{1}{4} \times \frac{4}{5} = \frac{4}{20} \]

### Step 4: Add the probabilities of both scenarios to find the total probability of contradiction.
\[ P(\text{Contradiction}) = P(A \text{ speaks truth} \text{ and } B \text{ lies}) + P(A \text{ lies} \text{ and } B \text{ speaks truth}) \]
\[ P(\text{Contradiction}) = \frac{3}{20} + \frac{4}{20} = \frac{7}{20} \]

### Final Answer:
The probability that A and B contradict each other in their statements is:
\[ \frac{7}{20} \]

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