Probability Calculation: Agreeing Witnesses & Truth

This problem requires us to calculate the probability that a statement is true, given that two independent witnesses, A and B, agree on the statement. We are provided with the individual probabilities of each witness speaking the truth.

Understanding the Probabilities

Let's define the events and their probabilities:

• Let $A_{true}$ be the event that witness A speaks the truth. We are given $P(A_{true}) = x$.
• Let $B_{true}$ be the event that witness B speaks the truth. We are given $P(B_{true}) = y$.

Since A and B are independent witnesses, the probability of specific combinations of them telling the truth or lying can be found by multiplying their individual probabilities.

Consequently, the probabilities of them lying are:

• Probability of A lying: $P(A_{false}) = 1 - P(A_{true}) = 1-x$.
• Probability of B lying: $P(B_{false}) = 1 - P(B_{true}) = 1-y$.

Events of Agreement

The core of the problem is the event where both witnesses agree. Agreement occurs in two distinct cases:

• Case 1: Both witnesses tell the truth. Since they are independent, the probability of this happening is $P(A_{true} \cap B_{true}) = P(A_{true}) \times P(B_{true}) = xy$.
• Case 2: Both witnesses lie. Similarly, due to independence, the probability of this is $P(A_{false} \cap B_{false}) = P(A_{false}) \times P(B_{false}) = (1-x)(1-y)$.

Let S be the event that witnesses A and B agree. Because Case 1 and Case 2 are mutually exclusive (they cannot happen at the same time), the total probability of agreement is the sum of the probabilities of these two cases:

$$P(S) = P(A_{true} \cap B_{true}) + P(A_{false} \cap B_{false}) = xy + (1-x)(1-y)$$

Conditional Probability Calculation

We need to find the probability that the statement is true, given that the witnesses agreed. Let T be the event that the statement is true. If the statement is true, it necessarily means that both witnesses must have spoken the truth ($A_{true} \cap B_{true}$).

We are looking for the conditional probability $P(T | S)$. The formula for conditional probability is:

$$P(T | S) = \frac{P(T \cap S)}{P(S)}$$

Now, let's consider the intersection event '$T \cap S$'. This event means that the statement is true AND the witnesses A and B agree. If the statement is true, both witnesses must have told the truth. If both witnesses told the truth, they automatically agree. Therefore, the event '$T \cap S$' is identical to the event that both witnesses tell the truth ($A_{true} \cap B_{true}$).

So, the probability of the intersection is $P(T \cap S) = P(A_{true} \cap B_{true}) = xy$.

Final Probability Derivation

Now, substitute the calculated probabilities into the conditional probability formula:

$$P(T | S) = \frac{P(T \cap S)}{P(S)} = \frac{xy}{xy + (1-x)(1-y)}$$

This final expression gives the probability that the statement is true, given the condition that both independent witnesses, A and B, agreed.