Question

the venn diagram below shows the 12 students in ms. aokis class. the diagram shows the memberships for the chess club and the science club. a student from the class is randomly selected. let a denote the event \the student is in the chess club.\ let b denote the event \the student is in the science club.\ the outcomes for the event a are listed in the circle on the left. the outcomes for the event b are listed in the circle on the right. note that yoko is outside the circles since she is not a member of either club. (a) find the probabilities of the events below. write each answer as a single fraction. p(a) = 9/12 p(b) = 6/12 p(a and b) = 4/12 p(b|a) = p(a)·p(b|a) =

Explanation:

Step1: Recall conditional - probability formula

The formula for conditional probability is $P(B|A)=\frac{P(A\cap B)}{P(A)}$.

Step2: Substitute known values

We know that $P(A)=\frac{9}{12}$ and $P(A\cap B)=\frac{4}{12}$.
Substituting into the formula: $P(B|A)=\frac{\frac{4}{12}}{\frac{9}{12}}$.

Step3: Simplify the fraction

When dividing by a fraction, we multiply by its reciprocal. So $P(B|A)=\frac{4}{12}\times\frac{12}{9}=\frac{4}{9}$.

Step4: Calculate $P(A)\cdot P(B|A)$

We know $P(A)=\frac{9}{12}$ and $P(B|A)=\frac{4}{9}$. Then $P(A)\cdot P(B|A)=\frac{9}{12}\times\frac{4}{9}=\frac{4}{12}=\frac{1}{3}$.

Answer:

$P(B|A)=\frac{4}{9}$
$P(A)\cdot P(B|A)=\frac{1}{3}$