Question

at a grocery store, the probability that a random customer will buy dairy is 0.63. the probability that the customer will buy produce is 0.57. the probability that the customer will buy either dairy or produce is 0.80. what is the probability that the customer will buy both dairy and produce?
a. 0.40
b. 0.45
c. 0.74
d. 0.86

Explanation:

Step1: Recall the formula for the probability of the union of two events

The formula for the probability of the union of two events \( A \) and \( B \) is \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \), where \( P(A \cup B) \) is the probability of either \( A \) or \( B \) occurring, \( P(A) \) and \( P(B) \) are the probabilities of \( A \) and \( B \) occurring respectively, and \( P(A \cap B) \) is the probability of both \( A \) and \( B \) occurring. Let \( A \) be the event that a customer buys dairy and \( B \) be the event that a customer buys produce. We know \( P(A) = 0.63 \), \( P(B) = 0.57 \), and \( P(A \cup B) = 0.80 \). We need to find \( P(A \cap B) \).

Step2: Rearrange the formula to solve for \( P(A \cap B) \)

Rearranging the formula \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \) gives \( P(A \cap B) = P(A) + P(B) - P(A \cup B) \).

Step3: Substitute the given values into the formula

Substitute \( P(A) = 0.63 \), \( P(B) = 0.57 \), and \( P(A \cup B) = 0.80 \) into the formula: \( P(A \cap B) = 0.63 + 0.57 - 0.80 \).

Step4: Calculate the result

First, add \( 0.63 \) and \( 0.57 \): \( 0.63 + 0.57 = 1.2 \). Then subtract \( 0.80 \) from \( 1.2 \): \( 1.2 - 0.80 = 0.40 \). Wait, that's not right. Wait, \( 0.63 + 0.57 = 1.2 \), then \( 1.2 - 0.80 = 0.4 \)? Wait, no, wait, let's recalculate. \( 0.63 + 0.57 = 1.2 \), then \( 1.2 - 0.80 = 0.4 \)? But wait, the options have 0.40 as option A, but let's check again. Wait, maybe I made a mistake. Wait, \( P(A \cap B)=P(A)+P(B)-P(A\cup B)=0.63 + 0.57-0.80 = 1.2 - 0.8 = 0.4 \)? But wait, the options: A is 0.40, B is 0.45, C is 0.74, D is 0.86. Wait, maybe I messed up the formula. Wait, no, the formula for the union of two events is correct. Wait, let's check the numbers again. \( 0.63 + 0.57 = 1.2 \), minus \( 0.80 \) is \( 0.4 \), which is 0.40, option A. But wait, maybe I made a mistake. Wait, no, the formula is \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \), so solving for \( P(A \cap B) \) gives \( P(A \cap B)=P(A)+P(B)-P(A \cup B) \). So substituting the values: \( 0.63 + 0.57 - 0.80 = 1.2 - 0.8 = 0.4 \), which is 0.40, option A. Wait, but let me check again. Wait, 0.63 + 0.57 is 1.2, minus 0.8 is 0.4. Yes, that's correct. So the probability that the customer buys both dairy and produce is 0.40. Wait, but the options have A as 0.40, so that's the answer. Wait, but maybe I made a mistake in the formula? No, the addition rule for probability is correct. The probability of either event A or event B occurring is equal to the probability of A plus the probability of B minus the probability of both A and B occurring. So that's the correct formula. So the calculation is correct. So the answer is 0.40, option A.

Answer:

A. 0.40