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In an experiment, the probability that event $A$ occurs is $\frac{1}{9}$ , the probability that event $B$ occurs is $\frac{3}{7}$ , and the probability that events $A$ and $B$ both occur is $\frac{1}{21}$ . $\newline$ 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{1}{9}

, the probability that event

B

occurs is

\frac{3}{7}

, and the probability that events

A

and

B

both occur is

\frac{1}{21}

.

\newline

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 $P(B)$ . $P(A) \times P(B) = \frac{1}{9} \times \frac{3}{7}$
Perform Multiplication: Now, do the multiplication. $\newline$ $(\frac{1}{9}) \times (\frac{3}{7}) = \frac{3}{63}$
Simplify Fraction: Simplify the fraction $\frac{3}{63}$ . $\frac{3}{63} = \frac{1}{21}$
Compare Results: Compare the result with $P(A \text{ and } B)$ . Since $P(A \text{ and } B)$ is also $\frac{1}{21}$ , this means $P(A) \times P(B) = P(A \text{ and } B)$ .
Confirm Independence: Since $P(A) \times P(B)$ equals $P(A \text{ and } B)$ , events $A$ and $B$ are independent.

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