36. Visits to the Doctor A National Ambulatory Medical Care Survey administered by the Centers for Disease Control found that the probability a randomly selected patient visited the doctor for a blood pressure check is 0.593. The probability a randomly selected patient visited the doctor for urinalysis is 0.064. Can we compute the probability of randomly selecting a patient who visited the doctor for a blood pressure check or urinalysis by adding these probabilities? Why or why not?

Verified step by step guidance

1. Step 1: Understand the problem context. We are given two probabilities: the probability a patient visited the doctor for a blood pressure check, \(P(B) = 0.593\), and the probability a patient visited for urinalysis, \(P(U) = 0.064\).
2. Step 2: Recall the rule for the probability of the union of two events. The probability that a patient visited for blood pressure check or urinalysis is given by the formula: \[P(B \cup U) = P(B) + P(U) - P(B \cap U)\] where \(P(B \cap U)\) is the probability that a patient visited for both reasons.
3. Step 3: Consider whether the two events are mutually exclusive. If the events are mutually exclusive (cannot happen at the same time), then \(P(B \cap U) = 0\), and the formula simplifies to \(P(B \cup U) = P(B) + P(U)\).
4. Step 4: Since the problem does not state that the events are mutually exclusive, we cannot assume \(P(B \cap U) = 0\). Therefore, simply adding the probabilities would overcount patients who visited for both reasons.
5. Step 5: Conclusion: Without knowing \(P(B \cap U)\), we cannot compute \(P(B \cup U)\) by just adding \(P(B)\) and \(P(U)\). We need information about the overlap to correctly calculate the combined probability.

Key Concepts

Addition Rule of Probability

The addition rule states that the probability of the union of two events equals the sum of their individual probabilities minus the probability of their intersection. This rule helps determine if simply adding probabilities is correct or if overlap must be considered.

Mutually Exclusive Events

Events are mutually exclusive if they cannot occur at the same time. If two events are mutually exclusive, their combined probability is the sum of their individual probabilities. Otherwise, overlap must be accounted for.

Intersection of Events

The intersection of two events represents the probability that both events occur simultaneously. When events are not mutually exclusive, this intersection must be subtracted to avoid double counting when calculating combined probabilities.