Question

in an experiment, the probability that event a occurs is $\frac{7}{8}$, the probability that event b occurs is $\frac{2}{3}$, and the probability that events a and b both occur is $\frac{5}{9}$. what is the probability that a occurs given that b occurs? simplify any fractions.

Explanation:

Step1: Recall conditional - probability formula

The formula for conditional probability is $P(A|B)=\frac{P(A\cap B)}{P(B)}$.

Step2: Identify given probabilities

We are given that $P(A)=\frac{7}{8}$, $P(B)=\frac{2}{3}$, and $P(A\cap B)=\frac{5}{9}$.

Step3: Substitute values into formula

Substitute $P(A\cap B)=\frac{5}{9}$ and $P(B)=\frac{2}{3}$ into the formula $P(A|B)=\frac{P(A\cap B)}{P(B)}$. So $P(A|B)=\frac{\frac{5}{9}}{\frac{2}{3}}$.

Step4: Simplify the fraction

When dividing by a fraction, we multiply by its reciprocal. So $P(A|B)=\frac{5}{9}\times\frac{3}{2}=\frac{5\times3}{9\times2}=\frac{15}{18}=\frac{5}{6}$.

Answer:

$\frac{5}{6}$