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In an experiment, the probability that event $A$ occurs is $\frac{3}{5}$ and the probability that event $B$ occurs is $\frac{1}{8}$ . 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{3}{5}

and the probability that event

B

occurs is

\frac{1}{8}

. If

A

and

B

are independent events, what is the probability that

A

and

B

both occur? Simplify any fractions.

Question Prompt: question_prompt: What is the probability that both event $A$ and event $B$ occur if $A$ and $B$ are independent events?
Multiplication of Probabilities: Since $A$ and $B$ are independent, multiply the probability of $A$ occurring by the probability of $B$ occurring to get the probability of both occurring: $P(A \text{ and } B) = P(A) \times P(B)$ . So, $P(A \text{ and } B) = \frac{3}{5} \times \frac{1}{8}$ .
Perform Multiplication: Perform the multiplication: $\frac{3}{5} \times \frac{1}{8} = \frac{3}{40}.$
Final Probability: There's no need to simplify further since $\frac{3}{40}$ is already in its simplest form.

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