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In an experiment, the probability that event $A$ occurs is $\frac{5}{8}$ , the probability that event $B$ occurs is $\frac{2}{9}$ , 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{5}{8}

, the probability that event

B

occurs is

\frac{2}{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: Calculate the probability of $A$ and $B$ both occurring if they were independent by multiplying $P(A)$ and $P(B)$ . $P(A) \times P(B) = \left(\frac{5}{8}\right) \times \left(\frac{2}{9}\right)$ = $\frac{10}{72}$ = $\frac{5}{36}$
Compare Probabilities: Compare the calculated probability with the given probability of $A$ and $B$ both occurring. $\newline$ Given $P(A \cap B) = \frac{5}{36}$ $\newline$ Calculated $P(A) \times P(B) = \frac{5}{36}$
Determine Independence: Determine if $A$ and $B$ are independent. Since $P(A \cap B) = P(A) \times P(B)$ , events $A$ and $B$ are independent.

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