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In an experiment, the probability that event $A$ occurs is $\frac{8}{9}$ , the probability that event $B$ occurs is $\frac{4}{9}$ , and the probability that events $A$ and $B$ both occur is $\frac{3}{7}$ . $\newline$ What is the probability that $A$ occurs given that $B$ occurs? $\newline$ Simplify any fractions. $\newline$ ____

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Find conditional probabilities

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Q. In an experiment, the probability that event

A

occurs is

\frac{8}{9}

, the probability that event

B

occurs is

\frac{4}{9}

, and the probability that events

A

and

B

both occur is

\frac{3}{7}

.

\newline

What is the probability that

A

occurs given that

B

occurs?

\newline

Simplify any fractions.

\newline

____

Use Conditional Probability Formula: To find the probability that $A$ occurs given that $B$ occurs, we use the formula for conditional probability: $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$ .
Identify Given Probabilities: We know $P(A \text{ and } B) = \frac{3}{7}$ and $P(B) = \frac{4}{9}$ . So, $P(A|B) = \frac{\frac{3}{7}}{\frac{4}{9}}$ .
Calculate $P(A|B)$ : To divide the fractions, we multiply by the reciprocal of the second fraction: $(\frac{3}{7}) \times (\frac{9}{4})$ .
Multiply Fractions: Now, multiply the numerators and the denominators: $(3 \times 9) / (7 \times 4)$ .
Simplify Fraction: This gives us $\frac{27}{28}$ as the simplified fraction.

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\newline

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