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In an experiment, the probability that event $A$ occurs is $\frac{4}{9}$ , the probability that event $B$ occurs is $\frac{2}{9}$ , 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{4}{9}

, the probability that event

B

occurs is

\frac{2}{9}

, 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.

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 $P(A \text{ and } B) = \frac{1}{9}$ and $P(B) = \frac{2}{9}$ . So, $P(A|B) = \frac{\frac{1}{9}}{\frac{2}{9}}$ .
Calculate Conditional Probability: Simplify the fraction by dividing $\frac{1}{9}$ by $\frac{2}{9}$ , which is the same as multiplying $\frac{1}{9}$ by the reciprocal of $\frac{2}{9}$ .
Simplify Fraction: $P(A|B) = \frac{1}{9} \times \frac{9}{2} = \frac{9}{18}$ .

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