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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 $\frac{2}{9}$ . If $A$ and $B$ are independent events, what is the probability that $A$ and $B$ both occur? 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{2}{9}

. If

A

and

B

are independent events, what is the probability that

A

and

B

both occur? Simplify any fractions.

Multiply probabilities of A and B: To find the probability of both A and B happening, multiply the probabilities of A and B since they're independent. So, $P(A \text{ and } B) = P(A) \times P(B)$ .
Calculate $P(A \text{ and } B)$ : $P(A)$ is $\frac{2}{9}$ and $P(B)$ is $\frac{2}{9}$ . So, $P(A \text{ and } B) = \frac{2}{9} \times \frac{2}{9}$ .
Perform multiplication: Now, do the multiplication: $\frac{2}{9} \times \frac{2}{9} = \frac{4}{81}$ .
Final result: No need to simplify, $\frac{4}{81}$ is already in its simplest form.

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