A grocer noticed that customers buy both pet and baby supplies with a probability of 0.12 . Considered separately, the probability that a customer buys pet supplies is P(A)=0.64, and the probability that a customer buys baby supplies is P(B)=0.25.
The correct answer and explanation is :
To solve this problem, we need to use the formula for the probability of the union of two events, A (buying pet supplies) and B (buying baby supplies). The problem provides the following probabilities:
- $P(A) = 0.64$ (the probability that a customer buys pet supplies)
- $P(B) = 0.25$ (the probability that a customer buys baby supplies)
- $P(A \cap B) = 0.12$ (the probability that a customer buys both pet and baby supplies)
We are asked to find the probability that a customer buys either pet supplies, baby supplies, or both. This can be expressed as the union of the two events:
$$
P(A \cup B) = P(A) + P(B) – P(A \cap B)
$$
Explanation of the Formula:
The formula above comes from the principle of inclusion-exclusion in probability theory. Here’s a breakdown:
- $P(A \cup B)$ is the probability that at least one of the two events happens (i.e., a customer buys either pet supplies, baby supplies, or both).
- $P(A)$ is the probability of buying pet supplies.
- $P(B)$ is the probability of buying baby supplies.
- $P(A \cap B)$ is the probability of buying both pet and baby supplies.
If we simply added $P(A)$ and $P(B)$, we would be double-counting the cases where customers buy both pet and baby supplies, so we subtract $P(A \cap B)$ to correct for this.
Calculation:
$$
P(A \cup B) = P(A) + P(B) – P(A \cap B)
$$
$$
P(A \cup B) = 0.64 + 0.25 – 0.12
$$
$$
P(A \cup B) = 0.77
$$
Conclusion:
The probability that a customer buys either pet supplies, baby supplies, or both is $0.77$, or 77%. This means there is a 77% chance that a customer will buy at least one of these items.
This calculation uses the principle of inclusion-exclusion, ensuring that we don’t double-count customers who purchase both pet and baby supplies. The result tells us that a substantial portion of customers buys either one or both types of products.