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In an experiment, the probability that event $A$ occurs is $\frac{8}{9}$ , the probability that event $B$ occurs is $\frac{4}{7}$ , and the probability that events $A$ and $B$ both occur is $\frac{32}{63}$ . 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{4}{7}

, and the probability that events

A

and

B

both occur is

\frac{32}{63}

. 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 $P(B)$ . $\newline$ $P(A) \times P(B) = \left(\frac{8}{9}\right) \times \left(\frac{4}{7}\right)$
Perform Multiplication: Now, do the multiplication. $\newline$ $(\frac{8}{9}) \times (\frac{4}{7}) = \frac{32}{63}$
Compare Product with $P(A \text{ and } B)$ : Compare the product of $P(A)$ and $P(B)$ with $P(A \text{ and } B)$ . Since $P(A \text{ and } B)$ is also $\frac{32}{63}$ , the product of $P(A)$ and $P(B)$ is equal to $P(A \text{ and } B)$ .
Confirm Independence of Events: Since $P(A) \times P(B) = P(A \text{ and } B)$ , events $A$ and $B$ are independent.

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