Question

in an experiment, the probability that event b occurs is $\frac{1}{3}$, and the probability that event a occurs given that event b occurs is $\frac{5}{6}$. 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.
We are given that $P(B)=\frac{1}{3}$ and $P(A|B)=\frac{5}{6}$.
We need to solve for $P(A\cap B)$.

Step2: Rearrange the formula

From $P(A|B)=\frac{P(A\cap B)}{P(B)}$, we can get $P(A\cap B)=P(A|B)\times P(B)$.

Step3: Substitute the given values

Substitute $P(A|B)=\frac{5}{6}$ and $P(B)=\frac{1}{3}$ into the formula $P(A\cap B)=P(A|B)\times P(B)$.
$P(A\cap B)=\frac{5}{6}\times\frac{1}{3}=\frac{5\times1}{6\times3}=\frac{5}{18}$.

Answer:

$\frac{5}{18}$