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In an experiment, the probability that event $A$ occurs is $\frac{5}{6}$ , the probability that event $B$ occurs is $\frac{3}{4}$ , and the probability that events $A$ and $B$ both occur is $\frac{5}{8}$ . 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{5}{6}

, the probability that event

B

occurs is

\frac{3}{4}

, and the probability that events

A

and

B

both occur is

\frac{5}{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 $P(A|B)$ : We know $P(A \text{ and } B) = \frac{5}{8}$ and $P(B) = \frac{3}{4}$ . So, $P(A|B) = \frac{\frac{5}{8}}{\frac{3}{4}}$ .
Multiply Fractions: To divide the fractions, we multiply by the reciprocal of the second fraction: $(\frac{5}{8}) \times (\frac{4}{3})$ .
Simplify Fraction: Now, multiply the numerators and denominators: $(5 \times 4) / (8 \times 3) = 20 / 24$ .
Simplify Fraction: Now, multiply the numerators and denominators: (5 \times 4) / (8 \times 3) = 20 / 24\. Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is \$4: $20 / 24 = (20/4) / (24/4) = 5 / 6$ .

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