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In an experiment, the probability that event $A$ occurs is $\frac{2}{3}$ , the probability that event $B$ occurs is $\frac{1}{3}$ , 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{2}{3}

, the probability that event

B

occurs is

\frac{1}{3}

, 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: $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$ .
Calculate $P(A \text{ and } B)$ and $P(B)$ : We know $P(A \text{ and } B) = \frac{1}{9}$ and $P(B) = \frac{1}{3}$ .
Calculate $P(A|B)$ : Now we calculate $P(A|B) = \frac{1}{9} / \frac{1}{3}$ .
Simplify the Fraction: Simplify the fraction by multiplying the numerator by the reciprocal of the denominator: $P(A|B) = \frac{1}{9} \times \frac{3}{1} = \frac{3}{9}$ .
Final Answer: Simplify $\frac{3}{9}$ to get the final answer: $P(A|B) = \frac{1}{3}$ .

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