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In an experiment, the probability that event $A$ occurs is $\frac{2}{5}$ , the probability that event $B$ occurs is $\frac{5}{9}$ , and the probability that events $A$ and $B$ both occur is $\frac{1}{3}$ . $\newline$ 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{2}{5}

, the probability that event

B

occurs is

\frac{5}{9}

, and the probability that events

A

and

B

both occur is

\frac{1}{3}

.

\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: $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$ .
Calculate $P(A|B)$ : We know $P(A \text{ and } B) = \frac{1}{3}$ and $P(B) = \frac{5}{9}$ . So, $P(A|B) = \frac{\frac{1}{3}}{\frac{5}{9}}$ .
Divide Fractions: To divide fractions, we multiply by the reciprocal of the divisor. So, $P(A|B) = \frac{1}{3} \times \frac{9}{5}$ .
Multiply Numerators and Denominators: Now, multiply the numerators and denominators: $(1 \times 9) / (3 \times 5) = \frac{9}{15}$ .
Simplify Fraction: Simplify the fraction $\frac{9}{15}$ by dividing both numerator and denominator by $3$ . So, $\frac{9}{15} = \frac{3}{5}$ .

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\newline

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\newline

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\newline

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