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In an experiment, the probability that event $A$ occurs is $\frac{1}{3}$ , the probability that event $B$ occurs is $\frac{5}{7}$ , and the probability that events $A$ and $B$ both occur is $\frac{1}{9}$ . What is the probability that $A$ occurs given that $B$ occurs? Simplify any fractions.

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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{5}{7}

, and the probability that events

A

and

B

both occur is

\frac{1}{9}

. What is the probability that

A

occurs given that

B

occurs? Simplify any fractions.

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 $P(A \text{ and } B) = \frac{1}{9}$ and $P(B) = \frac{5}{7}$ . So, $P(A|B) = \frac{\frac{1}{9}}{\frac{5}{7}}$ .
Multiply by Reciprocal: To divide the fractions, we multiply by the reciprocal of the second fraction: $(\frac{1}{9}) \times (\frac{7}{5})$ .
Multiply Numerators and Denominators: Now, multiply the numerators and denominators: $(1 \times 7) / (9 \times 5)$ .
Simplify the Result: This simplifies to $\frac{7}{45}$ . So, the probability that $A$ occurs given that $B$ occurs is $\frac{7}{45}$ .

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