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In an experiment, the probability that event $A$ occurs is $\frac{5}{9}$ , the probability that event $B$ occurs is $\frac{7}{8}$ , and the probability that events $A$ and $B$ both occur is $\frac{1}{2}$ . $\newline$ 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{5}{9}

, the probability that event

B

occurs is

\frac{7}{8}

, and the probability that events

A

and

B

both occur is

\frac{1}{2}

.

\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 $P(A \text{ and } B) = \frac{1}{2}$ and $P(B) = \frac{7}{8}$ . So, $P(A|B) = \frac{\frac{1}{2}}{\frac{7}{8}}$ .
Multiply by Reciprocal: To divide the fractions, we multiply by the reciprocal of the second fraction: $(\frac{1}{2}) \times (\frac{8}{7})$ .
Simplify the Fraction: Now, multiply the numerators and the denominators: $\frac{1 \times 8}{2 \times 7}$ .
Reduce the Fraction: This simplifies to $\frac{8}{14}$ , which can be further reduced to $\frac{4}{7}$ .

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\newline

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\newline

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\newline

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\newline

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\newline

Write your answer as a decimal rounded to the nearest thousandth.

\newline

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\newline

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