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In an experiment, the probability that event $A$ occurs is $\frac{6}{7}$ , the probability that event $B$ occurs is $\frac{3}{8}$ , and the probability that event $A$ occurs given that event $B$ occurs is $\frac{6}{7}$ . Are $A$ and $B$ independent events?

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Identify independent events

Full solution

Q. In an experiment, the probability that event

A

occurs is

\frac{6}{7}

, the probability that event

B

occurs is

\frac{3}{8}

, and the probability that event

A

occurs given that event

B

occurs is

\frac{6}{7}

. Are

A

and

B

independent events?

Define Independent Events: We are given: $\newline$ $P(A) = \frac{6}{7}$ $\newline$ $P(B) = \frac{3}{8}$ $\newline$ $P(A|B) = \frac{6}{7}$ $\newline$ Identify the definition of independent events. Two events $A$ and $B$ are independent if the occurrence of one does not affect the probability of the occurrence of the other, which mathematically means $P(A|B) = P(A)$ .
Compare Probabilities: Since we are given $P(A|B) = \frac{6}{7}$ , we compare it with $P(A)$ . We know: $P(A) = \frac{6}{7}$ $P(A|B) = \frac{6}{7}$ Check if $P(A|B)$ is equal to $P(A)$ . Since $P(A|B) = P(A)$ , this suggests that the occurrence of $B$ does not affect the probability of $A$ occurring.
Determine Independence: Determine if events $A$ and $B$ are independent or not based on the comparison. $\newline$ Since $P(A|B) = P(A)$ , by definition, events $A$ and $B$ are independent.

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