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In an experiment, the probability that event $A$ occurs is $\frac{1}{4}$ , the probability that event $B$ occurs is $\frac{5}{9}$ , and the probability that events $A$ and $B$ both occur is $\frac{5}{36}$ . $\newline$ 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}{4}

, the probability that event

B

occurs is

\frac{5}{9}

, and the probability that events

A

and

B

both occur is

\frac{5}{36}

.

\newline

Are

A

and

B

independent events?

\newline

Choices:

\newline

(A) yes

\newline

(B) no

Calculate Probability Product: Calculate $P(A) \times P(B)$ to see if it equals $P(A \cap B)$ . $\newline$ $P(A) = \frac{1}{4}, P(B) = \frac{5}{9}$ . $\newline$ $P(A) \times P(B) = \left(\frac{1}{4}\right) \times \left(\frac{5}{9}\right) = \frac{5}{36}.$
Compare Probability Values: Compare $P(A \cap B)$ with $P(A) \times P(B)$ . $\newline$ $P(A \cap B) = \frac{5}{36}$ , $P(A) \times P(B) = \frac{5}{36}$ . $\newline$ Since they are equal, $A$ and $B$ are independent.

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