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In an experiment, the probability that event $A$ occurs is $\frac{1}{6}$ and the probability that event $B$ occurs is $\frac{4}{7}$ . 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{1}{6}

and the probability that event

B

occurs is

\frac{4}{7}

. If

A

and

B

are independent events, what is the probability that

A

and

B

both occur?

\newline

Simplify any fractions.

Identify Independence: Since $A$ and $B$ are independent, the probability that both $A$ and $B$ occur is $P(A \text{ and } B) = P(A) \times P(B)$ . So we need to multiply the probabilities of $A$ and $B$ .
Calculate $P(A \text{ and } B)$ : $P(A)$ is $\frac{1}{6}$ and $P(B)$ is $\frac{4}{7}$ . Let's multiply these: $P(A \text{ and } B) = \frac{1}{6} \times \frac{4}{7}$ .
Multiply Probabilities: Multiplying the fractions gives us $P(A \text{ and } B) = \frac{1}{6} \times \frac{4}{7} = \frac{4}{42}$ .
Simplify Fraction: We can simplify $\frac{4}{42}$ by dividing both the numerator and the denominator by their greatest common divisor, which is $2$ .
Final Probability: After simplifying, we get $P(A \text{ and } B) = \frac{2}{21}$ .

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, and the probability that events

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B

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\text{yes}

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Question

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Question

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Question

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\newline

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Write your answer as a decimal rounded to the nearest thousandth.

\newline

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\newline

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