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In an experiment, the probability that event $A$ occurs is $\frac{7}{9}$ and the probability that event $B$ occurs is $\frac{3}{8}$ . 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{7}{9}

and the probability that event

B

occurs is

\frac{3}{8}

. If

A

and

B

are independent events, what is the probability that

A

and

B

both occur?

\newline

Simplify any fractions.

Calculate $P(A \text{ and } B)$ : Calculate $P(A \text{ and } B)$ for independent events $A$ and $B$ : $P(A \text{ and } B) = P(A) \times P(B)$ .
Substitute given probabilities: Substitute the given probabilities into the formula: $P(A \text{ and } B) = \frac{7}{9} \times \frac{3}{8}$ .
Perform the multiplication: Perform the multiplication: $P(A \text{ and } B) = \frac{7 \times 3}{9 \times 8}$ .
Simplify the multiplication: Simplify the multiplication: $P(A \text{ and } B) = \frac{21}{72}$ .
Reduce fraction: Reduce the fraction to its simplest form: $P(A \text{ and } B) = \frac{7}{24}$ .

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