Question

in an experiment, the probability that event a occurs is $\frac{1}{2}$, the probability that event b occurs is $\frac{3}{8}$, and the probability that events a and b both occur is $\frac{1}{4}$. 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 $P(A)=\frac{1}{2}$, $P(B)=\frac{3}{8}$, and $P(A\cap B)=\frac{1}{4}$.

Step3: Substitute values into formula

Substitute $P(A\cap B)=\frac{1}{4}$ and $P(B)=\frac{3}{8}$ into the formula $P(A|B)=\frac{P(A\cap B)}{P(B)}$, we get $P(A|B)=\frac{\frac{1}{4}}{\frac{3}{8}}$.

Step4: Simplify the fraction

To divide by a fraction, we multiply by its reciprocal. So $P(A|B)=\frac{1}{4}\times\frac{8}{3}=\frac{8}{12}=\frac{2}{3}$.

Answer:

$\frac{2}{3}$