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In an experiment, the probability that event $A$ occurs is $\frac{8}{9}$ and the probability that event $B$ occurs is $\frac{4}{9}$ . 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{8}{9}

and the probability that event

B

occurs is

\frac{4}{9}

. If

A

and

B

are independent events, what is the probability that

A

and

B

both occur?

\newline

Simplify any fractions.

Calculate Probability of A and B: Since $A$ and $B$ are independent, the probability that both $A$ and $B$ occur is $P(A \text{ and } B) = P(A) \times P(B)$ . So we need to multiply the probabilities of $A$ and $B$ .
Multiply Probabilities: $P(A)$ is $\frac{8}{9}$ and $P(B)$ is $\frac{4}{9}$ . Let's multiply them: $\frac{8}{9} \times \frac{4}{9}$ .
Final Probability Calculation: Multiplying the fractions gives us $\frac{32}{81}$ . This is the probability that both $A$ and $B$ occur.

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\newline

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\newline

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\newline

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