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In an experiment, the probability that event $A$ occurs is $\frac{8}{9}$ and the probability that event $B$ occurs is $\frac{1}{6}$ . 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{8}{9}

and the probability that event

B

occurs is

\frac{1}{6}

. 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 $P(A \text{ and } B)$ using the formula for independent events: $P(A \text{ and } B) = P(A) \times P(B)$ . So we got $P(A \text{ and } B) = \frac{8}{9} \times \frac{1}{6}$ .
Do the multiplication: Do the multiplication: $\frac{8}{9} \times \frac{1}{6} = \frac{8}{54}$ .
Simplify the fraction: Simplify the fraction $\frac{8}{54}$ by dividing both numerator and denominator by their greatest common divisor, which is $2$ . So, $\frac{8}{54}$ simplifies to $\frac{4}{27}$ .

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