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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 $\frac{1}{5}$ . 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{1}{5}

. If

A

and

B

are independent events, what is the probability that

A

and

B

both occur?

\newline

Simplify any fractions.

Multiply Probabilities: To find the probability that both $A$ and $B$ occur, we multiply the probabilities of $A$ and $B$ since they are independent. So, $P(A \text{ and } B) = P(A) \times P(B)$ .
Calculate $P(A \text{ and } B)$ : $P(A)$ is given as $\frac{5}{9}$ and $P(B)$ is given as $\frac{1}{5}$ . Now we calculate $P(A \text{ and } B) = \frac{5}{9} \times \frac{1}{5}$ .
Simplify Fraction: Multiplying the fractions, we get $P(A \text{ and } B) = \frac{5 \times 1}{9 \times 5}$ . This simplifies to $\frac{5}{45}$ .

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