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In an experiment, the probability that event $A$ occurs is $\frac{1}{7}$ , the probability that event $B$ occurs is $\frac{1}{9}$ , and the probability that events $A$ and $B$ both occur is $\frac{1}{63}$ . 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{1}{7}

, the probability that event

B

occurs is

\frac{1}{9}

, and the probability that events

A

and

B

both occur is

\frac{1}{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, let's find the product of $P(A)$ and $P(B)$ . $P(A) \times P(B) = \frac{1}{7} \times \frac{1}{9}$
Compare Product to Joint Probability: Now, calculate the product. $(\frac{1}{7}) \times (\frac{1}{9}) = \frac{1}{63}$
Confirm Independence: Next, compare this product to the probability of $A$ and $B$ occurring together, which is given as $\frac{1}{63}$ .
Conclusion: Since $\frac{1}{63}$ equals $\frac{1}{63}$ , the product of $P(A)$ and $P(B)$ is equal to $P(A \text{ and } B)$ .
Conclusion: Since $\frac{1}{63}$ equals $\frac{1}{63}$ , the product of $P(A)$ and $P(B)$ is equal to $P(A \text{ and } B)$ .Therefore, events $A$ and $B$ are independent because the product of their individual probabilities equals the probability of them occurring together.

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