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In an experiment, the probability that event $A$ occurs is $\frac{4}{9}$ , the probability that event $B$ occurs is $\frac{5}{8}$ , and the probability that events $A$ and $B$ both occur is $\frac{2}{9}$ . 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{4}{9}

, the probability that event

B

occurs is

\frac{5}{8}

, and the probability that events

A

and

B

both occur is

\frac{2}{9}

. What is the probability that

A

occurs given that

B

occurs? Simplify any fractions.

Identify $P(A|B)$ : We need to find $P(A|B)$ , which is the probability of $A$ given $B$ . The formula for conditional probability is $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$ .
Apply Conditional Probability Formula: We know $P(A \text{ and } B) = \frac{2}{9}$ and $P(B) = \frac{5}{8}$ . So, let's plug these values into the formula. $\newline$ $P(A|B) = \frac{\frac{2}{9}}{\frac{5}{8}}$
Substitute Values: To divide fractions, we multiply by the reciprocal of the second fraction. $P(A|B) = \left(\frac{2}{9}\right) \times \left(\frac{8}{5}\right)$
Simplify Fractions: Now, multiply the numerators and the denominators. $\newline$ $P(A|B) = \frac{2 \times 8}{9 \times 5}$
Calculate Final Probability: Simplify the multiplication. $P(A|B) = \frac{16}{45}$

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