Question

In an experiment, the probability that event B occurs is, and the probability that event A occurs given that event B occurs is 5 6 What is the probability that events A and B both occur? Simplify any fractions.

Answer 1

We are given the following information:

The probability that event B occurs is:

$P(B)=\frac{3}{5}$

And the probability that event A occurs given that event B occurs is:

$P(A|B)=\frac{5}{6}$

And we need to find the probability that both A and B occur.

To solve this problem, we have to use the conditional probability formula:

$P(A|B)=\frac{P(A\cap B)}{P(B)}$

Where

P(A|B) is the probability of A given that B occurred.

P(B) is the probability of B.

And P(A∩B) is the probability of A and B occuring.

Thus, we solve for P(A∩B) in the previous equation:

$P(A\cap B)=P(A|B)\cdot P(B)$

And substitute the known values:

$P(A\cap B)=\frac{5}{6}\cdot\frac{3}{5}$

We multiply the fractions and get the following result:

$\begin{gathered} P(A\cap B)=\frac{5\cdot3}{6\cdot5} \\ P(A\cap B)=\frac{15}{30} \end{gathered}$

Finally, we simplify the fraction by dividing both numbers in the fraction by 15:

$P(A\cap B)=\frac{1}{2}$

Answer: 1/2