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In an experiment, the probability that event $A$ occurs is $\frac{4}{9}$ and the probability that event $B$ occurs is $\frac{7}{8}$ . If $A$ and $B$ are independent events, what is the probability that $A$ and $B$ both occur? $\newline$ Simplify any fractions.

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Independence and conditional probability

Full solution

Q. In an experiment, the probability that event

A

occurs is

\frac{4}{9}

and the probability that event

B

occurs is

\frac{7}{8}

. If

A

and

B

are independent events, what is the probability that

A

and

B

both occur?

\newline

Simplify any fractions.

Multiply probabilities of A and B: To find the probability of both A and B occurring, multiply the probability of A by the probability of B since they're independent. So, $P(A \text{ and } B) = P(A) \times P(B)$ .
Calculate $P(A)$ and $P(B)$ : $P(A)$ is $\frac{4}{9}$ and $P(B)$ is $\frac{7}{8}$ . Now, let's multiply them: $P(A \text{ and } B) = \left(\frac{4}{9}\right) \times \left(\frac{7}{8}\right)$ .
Perform multiplication: Do the multiplication: $P(A \text{ and } B) = \frac{4}{9} \times \frac{7}{8} = \frac{28}{72}$ .

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