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In an experiment, the probability that event $A$ occurs is $\frac{6}{7}$ , the probability that event $B$ occurs is $\frac{1}{2}$ , and the probability that events $A$ and $B$ both occur is $\frac{4}{9}$ . 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{6}{7}

, the probability that event

B

occurs is

\frac{1}{2}

, and the probability that events

A

and

B

both occur is

\frac{4}{9}

. What is the probability that

A

occurs given that

B

occurs? Simplify any fractions.

Use Formula: To find the probability that A occurs given that B occurs, we use the formula $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$ .
Calculate Given Values: We know $P(A \text{ and } B) = \frac{4}{9}$ and $P(B) = \frac{1}{2}$ .
Calculate Probability: Now we calculate $P(A|B) = \frac{4}{9} / \frac{1}{2}$ .
Multiply by Reciprocal: To divide by a fraction, we multiply by its reciprocal. So, $P(A|B) = \frac{4}{9} \times \frac{2}{1}.$
Simplify Fraction: Multiplying the numerators and denominators, we get $P(A|B) = \frac{4 \times 2}{9 \times 1}$ .
Simplify Fraction: Multiplying the numerators and denominators, we get $P(A|B) = \frac{4 \times 2}{9 \times 1}$ . Simplifying, $P(A|B) = \frac{8}{9}$ .

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