Question

venn diagram below shows the 13 students in ms. woods class. diagram shows the memberships for the tennis club and the soccer club. a student from the class is randomly selected. let a denote the event “the student is in the tennis club.” let b denote the event “the student is in the soccer 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 reuben is outside the circles since he is not a member of either club. (a) find the probabilities of the events below. write each answer as a single fraction. p(a) = \frac{5}{13} p(b) = \frac{10}{13} p(a and b) = \frac{3}{13} p(a | b) = p(b)·p(a | b) = (b) select the probability that is equal to p(a and b). p(a|b)

Explanation:

Step1: Recall conditional - probability formula

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

Step2: Substitute known values

We know that $P(A\cap B)=\frac{3}{13}$ and $P(B)=\frac{10}{13}$. So, $P(A|B)=\frac{\frac{3}{13}}{\frac{10}{13}}$.

Step3: Simplify the fraction

When dividing by a fraction, we multiply by its reciprocal. So $P(A|B)=\frac{3}{13}\times\frac{13}{10}=\frac{3}{10}$.

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

Since $P(B)=\frac{10}{13}$ and $P(A|B)=\frac{3}{10}$, then $P(B)\cdot P(A|B)=\frac{10}{13}\times\frac{3}{10}=\frac{3}{13}$.

Answer:

$P(A|B)=\frac{3}{10}$
$P(B)\cdot P(A|B)=\frac{3}{13}$
(b) The probability that is equal to $P(A\ and\ B)$ is $P(B)\cdot P(A|B)$