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In an experiment, the probability that event $A$ occurs is $\frac{6}{7}$ and the probability that event $B$ occurs is $\frac{1}{9}$ . 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{6}{7}

and the probability that event

B

occurs is

\frac{1}{9}

. 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: $P(A \text{ and } B) = \left(\frac{6}{7}\right) \times \left(\frac{1}{9}\right)$ .
Perform multiplication: Perform the multiplication: $(\frac{6}{7}) \times (\frac{1}{9}) = \frac{6}{63}$ .
Simplify fraction: Simplify the fraction $\frac{6}{63}$ by dividing both numerator and denominator by the greatest common divisor, which is $9$ : $\frac{6}{63} \div \frac{9}{9} = \frac{6}{9} \div \frac{9}{9}.$

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