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In an experiment, the probability that event $A$ occurs is $\frac{4}{5}$ 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{4}{5}

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 of both $A$ and $B$ happening, we multiply their probabilities together since they're independent. So, we do $\frac{4}{5} \times \frac{1}{5}.$
Calculate Result: Now, let's multiply the numerators and denominators: $(4 \times 1) / (5 \times 5)$ equals $4/25$ .
Final Simplification: No need to simplify, $\frac{4}{25}$ is already in its simplest form.

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