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In an experiment, the probability that event $A$ occurs is $\frac{5}{9}$ 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{5}{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.

Identify Independence: $P(A \text{ and } B) = P(A) \times P(B)$ cuz they're independent, so let's multiply the probabilities.
Calculate Probabilities: $P(A \text{ and } B) = \frac{5}{9} \times \frac{4}{9}$ . Time to multiply the tops and the bottoms.
Multiply Probabilities: $P(A \text{ and } B) = \frac{20}{81}$ . That's the multiplication done, no need to simplify, looks good.

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