Question

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

Explanation:

Step1: Recall conditional - probability formula

The formula for conditional probability is $P(A|B)=\frac{P(A\cap B)}{P(B)}$, where $P(A|B)$ is the probability of event $A$ given event $B$, $P(A\cap B)$ is the probability of both $A$ and $B$ occurring, and $P(B)$ is the probability of event $B$ occurring.

Step2: Rearrange the formula to solve for $P(A\cap B)$

We can rewrite the formula as $P(A\cap B)=P(A|B)\times P(B)$.

Step3: Substitute the given values

We are given that $P(B)=\frac{2}{7}$ and $P(A|B)=\frac{2}{5}$. Substituting these values into the formula, we get $P(A\cap B)=\frac{2}{5}\times\frac{2}{7}$.

Step4: Calculate the product of the fractions

$\frac{2}{5}\times\frac{2}{7}=\frac{2\times2}{5\times7}=\frac{4}{35}$.

Answer:

$\frac{4}{35}$