A speaks truth in 60% cases and B speaks truth in 70% cases. The probability that they will say the same thing while describing a single event is:

Options

A. a. `0.36`
B. b. `0.54`
C. c. `0.48`
D. d. `0.20`

Answer

To find the probability that A and B will say the same thing while describing a single event, we need to consider two scenarios:

1. Both A and B tell the truth.
2. Both A and B tell a lie.

Let's denote:
- P(A tells the truth) = 0.6 (60%)
- P(A tells a lie) = 1 - P(A tells the truth) = 1 - 0.6 = 0.4 (40%)
- P(B tells the truth) = 0.7 (70%)
- P(B tells a lie) = 1 - P(B tells the truth) = 1 - 0.7 = 0.3 (30%)

### Step 1: Calculate the probability that both A and B tell the truth.
The probability that both A and B tell the truth can be calculated by multiplying their individual probabilities of telling the truth:
\[ P(A \text{ tells the truth}) \times P(B \text{ tells the truth}) = 0.6 \times 0.7 = 0.42 \]

### Step 2: Calculate the probability that both A and B tell a lie.
Similarly, the probability that both A and B tell a lie can be calculated by multiplying their individual probabilities of telling a lie:
\[ P(A \text{ tells a lie}) \times P(B \text{ tells a lie}) = 0.4 \times 0.3 = 0.12 \]

### Step 3: Add the probabilities from Step 1 and Step 2.
To find the total probability that A and B say the same thing (either both tell the truth or both tell a lie), we add the probabilities from Step 1 and Step 2:
\[ P(\text{same}) = P(A \text{ tells the truth}) \times P(B \text{ tells the truth}) + P(A \text{ tells a lie}) \times P(B \text{ tells a lie}) \]
\[ P(\text{same}) = 0.42 + 0.12 = 0.54 \]

### Final Answer:
The probability that A and B will say the same thing while describing a single event is **0.54**.

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