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In an experiment, the probability that event $A$ occurs is $\frac{4}{9}$ and the probability that event $B$ occurs is $\frac{1}{7}$ . 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{4}{9}

and the probability that event

B

occurs is

\frac{1}{7}

. If

A

and

B

are independent events, what is the probability that

A

and

B

both occur? Simplify any fractions.

Given Probabilities: $P(A)$ is $\frac{4}{9}$ and $P(B)$ is $\frac{1}{7}$ . Since $A$ and $B$ are independent, $P(A \text{ and } B) = P(A) \times P(B)$ .
Multiplying Probabilities: Now, let's multiply the probabilities: $\frac{4}{9} \times \frac{1}{7}.$
Calculation: Doing the multiplication: $\frac{4}{9} \times \frac{1}{7} = \frac{4}{63}$ .
Final Result: No need to simplify, $\frac{4}{63}$ is already in simplest form.

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