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In an experiment, the probability that event $A$ occurs is $\frac{8}{9}$ , the probability that event $B$ occurs is $\frac{6}{7}$ , and the probability that events $A$ and $B$ both occur is $\frac{4}{5}$ . 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{8}{9}

, the probability that event

B

occurs is

\frac{6}{7}

, and the probability that events

A

and

B

both occur is

\frac{4}{5}

. 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 $P(A|B)$ : We know $P(A \text{ and } B) = \frac{4}{5}$ and $P(B) = \frac{6}{7}$ . So, $P(A|B) = \frac{\frac{4}{5}}{\frac{6}{7}}$ .
Multiply Fractions: To divide the fractions, we multiply by the reciprocal of the second fraction: $(\frac{4}{5}) \times (\frac{7}{6})$ .
Simplify Fraction: Now, multiply the numerators and the denominators: $(4 \times 7) / (5 \times 6) = 28 / 30$ .
Simplify Fraction: Now, multiply the numerators and the denominators: $(4 \times 7) / (5 \times 6) = 28 / 30$ .Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is $2$ : $28 / 30 = 14 / 15$ .

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