A speaks the truth in 70 percent cases and B in 80 percent. The probability that they will contradict each. other when describing a single event is

Options

A. 0.36
B. 0.38
C. 0.4
D. 0.42

Answer

To find the probability that A and B will contradict each other when describing a single event, we can break down the problem into two cases:

### Step 1: Define the probabilities
- Let \( P(A \text{ tells the truth}) = 0.7 \)
- Therefore, \( P(A \text{ lies}) = 1 - P(A \text{ tells the truth}) = 1 - 0.7 = 0.3 \)
- Let \( P(B \text{ tells the truth}) = 0.8 \)
- Therefore, \( P(B \text{ lies}) = 1 - P(B \text{ tells the truth}) = 1 - 0.8 = 0.2 \)

### Step 2: Calculate the probability of the first case
In the first case, A is lying and B is telling the truth:
- Probability that A lies and B tells the truth is:
\[ P(A \text{ lies}) \times P(B \text{ tells the truth}) = 0.3 \times 0.8 = 0.24 \]

### Step 3: Calculate the probability of the second case
In the second case, A is telling the truth and B is lying:
- Probability that A tells the truth and B lies is:
\[ P(A \text{ tells the truth}) \times P(B \text{ lies}) = 0.7 \times 0.2 = 0.14 \]

### Step 4: Add the probabilities of both cases
To find the total probability that A and B contradict each other, we add the probabilities of the two cases:
\[ P(\text{contradiction}) = P(A \text{ lies and } B \text{ tells the truth}) + P(A \text{ tells the truth and } B \text{ lies}) \]
\[ P(\text{contradiction}) = 0.24 + 0.14 = 0.38 \]

### Final Answer
The probability that A and B will contradict each other when describing a single event is \( 0.38 \).

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