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In an experiment, the probability that event $A$ occurs is $\frac{1}{3}$ , the probability that event $B$ occurs is $\frac{2}{3}$ , and the probability that events $A$ and $B$ both occur is $\frac{1}{9}$ . 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{1}{3}

, the probability that event

B

occurs is

\frac{2}{3}

, and the probability that events

A

and

B

both occur is

\frac{1}{9}

. Are

A

and

B

independent events?

\newline

Choices:

\newline

(A)yes

\newline

(B)no

Check for Independence: 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 $P(A) \times P(B)$ : Calculate $P(A) \times P(B)$ : $\frac{1}{3} \times \frac{2}{3} = \frac{2}{9}$ .
Compare Probabilities: Compare $P(A) \times P(B)$ with $P(A \text{ and } B)$ : $P(A \text{ and } B)$ is given as $\frac{1}{9}$ , but we calculated $P(A) \times P(B)$ as $\frac{2}{9}$ .
Conclusion: Since $P(A \text{ and } B)$ is not equal to $P(A) \times P(B)$ , events $A$ and $B$ are not independent.

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