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In an experiment, the probability that event $A$ occurs is $\frac{4}{9}$ , the probability that event $B$ occurs is $\frac{3}{4}$ , and the probability that events $A$ and $B$ both occur is $\frac{2}{9}$ . 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{4}{9}

, the probability that event

B

occurs is

\frac{3}{4}

, and the probability that events

A

and

B

both occur is

\frac{2}{9}

. Are

A

and

B

independent events?

\newline

Choices:

\newline

(A) yes

\newline

(B) no

Check Independence Criteria: 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 ( $P(A) \times P(B)$ ).
Calculate Product of Probabilities: First, calculate the product of $P(A)$ and $P(B)$ . $P(A) \times P(B) = \left(\frac{4}{9}\right) \times \left(\frac{3}{4}\right)$
Multiply Probabilities: Now, do the multiplication. $\newline$ $(\frac{4}{9}) \times (\frac{3}{4}) = \frac{12}{36}$
Simplify Fraction: Simplify the fraction $\frac{12}{36}$ to its lowest terms. $\newline$ $\frac{12}{36} = \frac{1}{3}$
Compare Probabilities: Compare $P(A \text{ and } B)$ with $P(A) \times P(B)$ . $\newline$ $P(A \text{ and } B) = \frac{2}{9}$ $\newline$ $P(A) \times P(B) = \frac{1}{3}$
Determine Independence: Since $\frac{2}{9}$ is not equal to $\frac{1}{3}$ , events $A$ and $B$ are not independent.

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