Coupon Usage Probability

Solution & Explanation

To solve the problem, we need to determine the probability that both riders use their coupons given that at least one of them does, denoted as $P(B | A)$. This is a classic application of conditional probability and can be solved using Bayes' theorem.

Step 1: Define the Events

• Event A: At least one rider uses the coupon.
• Event B: Both riders use the coupon.

Step 2: Understand the Problem

We are given:

• The probability that any one rider uses their coupon is $P$.

We need to find $P(B | A)$, the probability that both riders use the coupon given that at least one does.

Step 3: Use Bayes' Theorem

Bayes' theorem states:

$P(B | A) = \frac{P(A | B) \cdot P(B)}{P(A)}$

• $P(A | B)$: This is the probability that at least one rider uses the coupon given that both have used it. Clearly, this is 1, because if both use the coupon, then at least one has used it.

• $P(B)$: This is the probability that both riders use the coupon. Since the events are independent, this is given by:

  $P(B) = P^2$

• $P(A)$: This is the probability that at least one rider uses the coupon. This can be calculated using the complement rule:

  $P(A) = 1 - P(\text{neither rider uses the coupon})$

  The probability that neither rider uses the coupon is $(1 - P)^2$. Therefore:

  $P(A) = 1 - (1 - P)^2 = 2P - P^2$

Step 4: Calculate $P(B | A)$

Substituting these into Bayes' theorem:

$P(B | A) = \frac{1 \cdot P^2}{2P - P^2} = \frac{P^2}{2P - P^2} = \frac{P}{2 - P}$

Conclusion

The probability that both riders use their coupons given that at least one of them does is $\frac{P}{2 - P}$. This solution leverages the principles of binomial probability and conditional probability, demonstrating the utility of these concepts in practical decision-making scenarios.