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In an experiment, the probability that event $A$ occurs is $\frac{2}{9}$ and the probability that event $B$ occurs is $\frac{8}{9}$ . 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{2}{9}

and the probability that event

B

occurs is

\frac{8}{9}

. 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 that both A and B occur, multiply the probability of A by the probability of B since they are independent. $\newline$ $P(A \text{ and } B) = P(A) \times P(B) = \left(\frac{2}{9}\right) \times \left(\frac{8}{9}\right)$ .
Calculate the product: Now, do the multiplication: $(\frac{2}{9}) \times (\frac{8}{9}) = \frac{16}{81}$ .
Final probability: No need to simplify further, $\frac{16}{81}$ is already in its simplest form.

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