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In an experiment, the probability that event $A$ occurs is $\frac{5}{8}$ , the probability that event $B$ occurs is $\frac{2}{7}$ , and the probability that events $A$ and $B$ both occur is $\frac{5}{28}$ . $\newline$ 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{5}{8}

, the probability that event

B

occurs is

\frac{2}{7}

, and the probability that events

A

and

B

both occur is

\frac{5}{28}

.

\newline

Are

A

and

B

independent events?

\newline

Choices:

\newline

(A) yes

\newline

(B) no

Calculate Probability: Calculate the probability of $A$ and $B$ occurring together if they are independent. $P(A) = \frac{5}{8}$ , $P(B) = \frac{2}{7}$ . $P(A) \times P(B) = \left(\frac{5}{8}\right) \times \left(\frac{2}{7}\right) = \frac{10}{56} = \frac{5}{28}$ .
Compare with Joint Probability: Compare the calculated probability with the given joint probability. $\newline$ $P(A \cap B) = \frac{5}{28}$ . $\newline$ Since $P(A) \times P(B) = P(A \cap B)$ , events A and B are independent.
State Conclusion: State the conclusion based on the comparison. $\newline$ Since the calculated probability matches the given joint probability, $A$ and $B$ are independent events.

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