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Chapter 4: Problem 51

Suppose that \(P(A)=.4\) and \(P(A \cap B)=.12\). a. Find \(P(B \mid A)\). b. Are events \(A\) and \(B\) mutually exclusive? c. If \(P(B)=.3,\) are events \(A\) and \(B\) independent?

Short Answer

Expert verified
Answer: The events A and B are not mutually exclusive, but they are independent.

Step by step solution

01

Finding the conditional probability

The conditional probability of event B given event A, denoted by \(P(B \mid A)\), is defined as: $$P(B \mid A) = \frac{P(A \cap B)}{P(A)}$$ We have both \(P(A \cap B) = 0.12\) and \(P(A) = 0.40\), so we can calculate the conditional probability: $$P(B \mid A) = \frac{0.12}{0.40} = 0.3$$
02

Determining if events A and B are mutually exclusive

Two events A and B are considered mutually exclusive if and only if their intersection is the empty set, meaning they cannot both occur at the same time. In terms of probability, two events are mutually exclusive if \(P(A \cap B) = 0\). The given probability of \(P(A \cap B) = 0.12\), which means events A and B are not mutually exclusive, as their intersection probability is not equal to zero.
03

Determining if events A and B are independent given their individual probabilities

Two events A and B are considered independent if and only if the occurrence of one event does not affect the probability of the other event occurring. Mathematically, this means: $$P(A \cap B) = P(A) \cdot P(B)$$ We are given that \(P(A) = 0.4\), \(P(A \cap B)= 0.12\) and \(P(B) = 0.3\). To check if events A and B are independent, we can compare the values of \(P(A) \cdot P(B)\) and \(P(A \cap B)\): $$P(A) \cdot P(B) = 0.4 \cdot 0.3 = 0.12$$ Since \(P(A \cap B) = P(A) \cdot P(B)\), events A and B are independent. To summarize: a. \(P(B \mid A) = 0.3\) b. Events A and B are not mutually exclusive. c. Events A and B are independent.

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