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In an experiment, the probability that event $A$ occurs is $\frac{8}{9}$ , the probability that event $B$ occurs is $\frac{5}{6}$ , and the probability that events $A$ and $B$ both occur is $\frac{20}{27}$ . Are $A$ and $B$ independent events? $\newline$ Choices: $\newline$ (A) yes $\newline$ (B) no

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Identify independent events

Full solution

Q. In an experiment, the probability that event

A

occurs is

\frac{8}{9}

, the probability that event

B

occurs is

\frac{5}{6}

, and the probability that events

A

and

B

both occur is

\frac{20}{27}

. Are

A

and

B

independent events?

\newline

Choices:

\newline

(A) yes

\newline

(B) no

Calculate Probabilities: To check if events $A$ and $B$ are independent, we need to see if the probability of $A$ and $B$ occurring together ( $P(A \text{ and } B)$ ) is equal to the product of their individual probabilities ( $P(A) \times P(B)$ ).
Multiply Probabilities: First, calculate $P(A) \times P(B)$ . $\newline$ $P(A) = \frac{8}{9}$ and $P(B) = \frac{5}{6}$ . $\newline$ So, $P(A) \times P(B) = \left(\frac{8}{9}\right) \times \left(\frac{5}{6}\right)$ .
Simplify Fraction: Now, do the multiplication. $\newline$ $(\frac{8}{9}) \times (\frac{5}{6}) = \frac{40}{54}$ . $\newline$ But wait, we can simplify this fraction by dividing both numerator and denominator by $2$ . $\newline$ So, $\frac{40}{54}$ simplifies to $\frac{20}{27}$ .
Compare Probabilities: Next, compare $P(A) \times P(B)$ with $P(A \text{ and } B)$ . We found $P(A) \times P(B) = \frac{20}{27}$ , and we're given $P(A \text{ and } B) = \frac{20}{27}$ .
Determine Independence: Since $P(A) \times P(B)$ is equal to $P(A \text{ and } B)$ , events $A$ and $B$ are independent.

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