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In an experiment, the probability that event $A$ occurs is $\frac{4}{9}$ , the probability that event $B$ occurs is $\frac{1}{2}$ , and the probability that events $A$ and $B$ both occur is $\frac{1}{9}$ . $\newline$ 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{1}{2}

, and the probability that events

A

and

B

both occur is

\frac{1}{9}

.

\newline

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 $P(A) \times P(B)$ : $\frac{4}{9} \times \frac{1}{2}$ .
Compare Probabilities: $P(A) \times P(B) = \frac{4}{9} \times \frac{1}{2} = \frac{4}{18} = \frac{2}{9}.$
Final Conclusion: Compare $P(A \text{ and } B)$ with $P(A) \times P(B)$ : $P(A \text{ and } B)$ is $\frac{1}{9}$ , while $P(A) \times P(B)$ is $\frac{2}{9}$ .
Final Conclusion: Compare $P(A \text{ and } B)$ with $P(A) \times P(B)$ : $P(A \text{ and } B)$ is $\frac{1}{9}$ , while $P(A) \times P(B)$ is $\frac{2}{9}$ .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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