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Chapter 4: Problem 49

Suppose that \(P(A)=.4\) and \(P(B)=.2\). If events \(A\) and \(B\) are independent, find these probabilities: a. \(P(A \cap B)\) b. \(P(A \cup B)\)

Short Answer

Expert verified
Answer: The probability of the intersection (A and B occurring together) is 0.08, and the probability of the union (either A occurs, B occurs, or both occur) is 0.52.

Step by step solution

01

Determine the intersection probability of independent events

Since the events A and B are independent, we can find the probability of their intersection (A and B occurring together) by multiplying their individual probabilities: \(P(A \cap B) = P(A) \times P(B)\)
02

Calculate the probability of A and B together

Using the given probabilities, we can calculate \(P(A \cap B)\): \(P(A \cap B) = P(A) \times P(B) = 0.4 \times 0.2 = 0.08\) So, the probability of A and B occurring together is 0.08.
03

Determine the union probability of independent events

To find the probability of the union of A and B (either A occurs, B occurs, or both occur), we can use the formula for the union of independent events: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
04

Calculate the probability of A or B or both occurring

We already found the probability of A and B occurring together in Step 2. So, we can substitute the values into the formula: \(P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.4 + 0.2 - 0.08 = 0.52\) So, the probability of either A occurring, B occurring, or both occurring is 0.52. Now, we have the probabilities of the intersection and the union of A and B: a. \(P(A \cap B) = 0.08\) b. \(P(A \cup B) = 0.52\)

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