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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 $\frac{7}{9}$ . 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{7}{9}

. If

A

and

B

are independent events, what is the probability that

A

and

B

both occur?

\newline

Simplify any fractions.

Multiply Probabilities: To find the probability of both $A$ and $B$ happening, we multiply their probabilities together since they're independent. So, we do $\frac{6}{7} \times \frac{7}{9}.$
Calculate Result: Now, let's multiply the numerators and denominators: $(6 \times 7) / (7 \times 9) = 42 / 63$ .
Simplify Fraction: We can simplify $\frac{42}{63}$ by dividing both the numerator and the denominator by their greatest common divisor, which is $21$ . So, $42 \div 21 = 2$ and $63 \div 21 = 3$ .
Final Probability: After simplifying, we get the fraction $\frac{2}{3}$ . So, the probability that both $A$ and $B$ occur is $\frac{2}{3}$ .

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