Question

in an experiment, the probability that event b occurs is $\frac{1}{4}$, and the probability that event a occurs given that event b occurs is $\frac{3}{5}$. what is the probability that events a and b both occur? simplify any fractions.

Explanation:

Step1: Recall the formula for conditional probability

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 $P(B)=\frac{1}{4}$ and $P(A|B)=\frac{3}{5}$. So, $P(A\cap B)=\frac{3}{5}\times\frac{1}{4}$.

Step4: Calculate the product

$\frac{3}{5}\times\frac{1}{4}=\frac{3\times1}{5\times4}=\frac{3}{20}$.

Answer:

$\frac{3}{20}$