Question

in an experiment, the probability that event a occurs is $\frac{7}{9}$, the probability that event b occurs is $\frac{4}{9}$, and the probability that events a and b both occur is $\frac{2}{5}$. 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 $P(A)=\frac{7}{9}$, $P(B)=\frac{4}{9}$, and $P(A\cap B)=\frac{2}{5}$.

Step3: Substitute values into formula

Substitute $P(A\cap B)=\frac{2}{5}$ and $P(B)=\frac{4}{9}$ into the formula $P(A|B)=\frac{P(A\cap B)}{P(B)}$. So $P(A|B)=\frac{\frac{2}{5}}{\frac{4}{9}}$.

Step4: Simplify the fraction

When dividing by a fraction, we multiply by its reciprocal. So $P(A|B)=\frac{2}{5}\times\frac{9}{4}=\frac{18}{20}=\frac{9}{10}$.

Answer:

$\frac{9}{10}$