Question

in an experiment, the probability that event a occurs is $\frac{1}{3}$, the probability that event b occurs is $\frac{2}{9}$, and the probability that events a and b both occur is $\frac{1}{7}$. 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: Substitute given values

We are given that $P(A)=\frac{1}{3}$, $P(B)=\frac{2}{9}$, and $P(A\cap B)=\frac{1}{7}$. Substituting $P(A\cap B)=\frac{1}{7}$ and $P(B)=\frac{2}{9}$ into the formula $P(A|B)=\frac{P(A\cap B)}{P(B)}$, we get $P(A|B)=\frac{\frac{1}{7}}{\frac{2}{9}}$.

Step3: Simplify the fraction

When dividing by a fraction, we multiply by its reciprocal. So $P(A|B)=\frac{1}{7}\times\frac{9}{2}=\frac{9}{14}$.

Answer:

$\frac{9}{14}$