X speaks truth in 60 % and Y in 50 % of the cases . The probability that they contradict each other while narrating the same fact is a) 1/4 b)1/3 c)1/2 d)2/3

Options

A. `1/4`
B. `1/3`
C. `1/2`
D. `2/3`

Answer

To solve the problem step by step, we will calculate the probability that X and Y contradict each other while narrating the same fact.

### Step 1: Define the events
Let:
- Event A: X speaks the truth.
- Event B: Y speaks the truth.

### Step 2: Determine the probabilities
From the problem statement:
- The probability that X speaks the truth, \( P(A) = 60\% = \frac{60}{100} = \frac{3}{5} \).
- The probability that Y speaks the truth, \( P(B) = 50\% = \frac{50}{100} = \frac{1}{2} \).

### Step 3: Calculate the probabilities of lying
- The probability that X lies (complement of A), \( P(A') = 1 - P(A) = 1 - \frac{3}{5} = \frac{2}{5} \).
- The probability that Y lies (complement of B), \( P(B') = 1 - P(B) = 1 - \frac{1}{2} = \frac{1}{2} \).

### Step 4: Determine the conditions for contradiction
X and Y will contradict each other if:
1. X speaks the truth and Y lies.
2. X lies and Y speaks the truth.

### Step 5: Calculate the probability of contradiction
The probability that they contradict each other can be calculated as:
\[ P(\text{Contradiction}) = P(A) \cdot P(B') + P(A') \cdot P(B) \]
Substituting the values we found:
\[ P(\text{Contradiction}) = P(A) \cdot P(B') + P(A') \cdot P(B) = \left(\frac{3}{5} \cdot \frac{1}{2}\right) + \left(\frac{2}{5} \cdot \frac{1}{2}\right) \]

### Step 6: Simplify the expression
Calculating each term:
\[ = \frac{3}{10} + \frac{2}{10} = \frac{5}{10} = \frac{1}{2} \]

### Conclusion
The probability that X and Y contradict each other while narrating the same fact is \( \frac{1}{2} \).

### Final Answer
Thus, the answer is \( \frac{1}{2} \) or option (c).

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