Understanding the Probability Relationship

This question asks for the minimum possible value of the sum of the probabilities of two events, \(P(A) + P(B)\), given certain conditions on their union and intersection.

We can use the fundamental formula relating the probabilities of two events A and B:

\(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)

By rearranging this formula, we can express the sum \(P(A) + P(B)\) in terms of the union and intersection:

\(P(A) + P(B) = P(A \cup B) + P(A \cap B)\)

Applying Constraints to Find Minimum Value

The problem provides the following constraints:

• The probability of the union of A and B is at least 0.75: \(P(A \cup B) \ge 0.75\).
• The probability of the intersection of A and B is between 0.125 and 0.375 (inclusive): \(0.125 \le P(A \cap B) \le 0.375\).

To find the minimum value of \(P(A) + P(B)\), we need to find the minimum value of the sum \(P(A \cup B) + P(A \cap B)\). This occurs when we use the minimum possible values for both \(P(A \cup B)\) and \(P(A \cap B)\) allowed by the constraints.

• The minimum value for \(P(A \cup B)\) is \(0.75\).
• The minimum value for \(P(A \cap B)\) is \(0.125\).

Step-by-Step Calculation

Now, substitute these minimum values into the rearranged formula:

Minimum value of \(P(A) + P(B) = \min(P(A \cup B)) + \min(P(A \cap B))\)

Minimum value of \(P(A) + P(B) = 0.75 + 0.125\)

Minimum value of \(P(A) + P(B) = 0.875\)

Conclusion

Therefore, the minimum value of \(P(A) + P(B)\) under the given conditions is \(0.875\).