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In an experiment, the probability that event $A$ occurs is $\frac{1}{3}$ , the probability that event $B$ occurs is $\frac{7}{9}$ , and the probability that events $A$ and $B$ both occur is $\frac{2}{7}$ . $\newline$ What is the probability that $A$ occurs given that $B$ occurs? $\newline$ Simplify any fractions. $\newline$ ____ $\newline$

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Find conditional probabilities

Full solution

Q. In an experiment, the probability that event

A

occurs is

\frac{1}{3}

, the probability that event

B

occurs is

\frac{7}{9}

, and the probability that events

A

and

B

both occur is

\frac{2}{7}

.

\newline

What is the probability that

A

occurs given that

B

occurs?

\newline

Simplify any fractions.

\newline

____

\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 $P(A \text{ and } B) = \frac{2}{7}$ and $P(B) = \frac{7}{9}$ . So, $P(A|B) = \frac{\frac{2}{7}}{\frac{7}{9}}$ .
Calculate Conditional Probability: To divide the fractions, we multiply by the reciprocal of the second fraction: $(\frac{2}{7}) \times (\frac{9}{7})$ .
Simplify Fraction: Multiplying the numerators and denominators, we get $(2 \times 9) / (7 \times 7) = 18/49$ .

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\newline

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\newline

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\newline

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\newline

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