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In an experiment, the probability that event $A$ occurs is $\frac{2}{3}$ , the probability that event $B$ occurs is $\frac{5}{8}$ , and the probability that events $A$ and $B$ both occur is $\frac{3}{8}$ . What is the probability that $A$ occurs given that $B$ occurs? Simplify any fractions.

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Find conditional probabilities

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Q. In an experiment, the probability that event

A

occurs is

\frac{2}{3}

, the probability that event

B

occurs is

\frac{5}{8}

, and the probability that events

A

and

B

both occur is

\frac{3}{8}

. What is the probability that

A

occurs given that

B

occurs? Simplify any fractions.

Use Conditional Probability Formula: To find the probability that $A$ occurs given that $B$ occurs, we use the formula for conditional probability: $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$ .
Calculate $P(A \text{ and } B)$ : We know $P(A \text{ and } B) = \frac{3}{8}$ and $P(B) = \frac{5}{8}$ .
Calculate $P(A|B)$ : Now we calculate $P(A|B) = \frac{3}{8} / \frac{5}{8}$ .
Simplify Fraction: Simplify the fraction by multiplying the numerator and the denominator by the reciprocal of the denominator: $(\frac{3}{8}) \times (\frac{8}{5})$ .
Final Probability Calculation: The eights cancel out, and we are left with $3 \times \frac{1}{5}$ .
Final Probability Calculation: The eights cancel out, and we are left with $3 \times \frac{1}{5}$ .So, $P(A|B) = \frac{3}{5}$ .

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