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In an experiment, the probability that event $A$ occurs is $\frac{1}{2}$ , the probability that event $B$ occurs is $\frac{4}{7}$ , and the probability that events $A$ and $B$ both occur is $\frac{1}{7}$ . 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}{2}

, the probability that event

B

occurs is

\frac{4}{7}

, and the probability that events

A

and

B

both occur is

\frac{1}{7}

. Are

A

and

B

independent events?

\newline

Choices:

\newline

(A) yes

\newline

(B) no

Calculate product of probabilities: 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 ( $P(A) \times P(B)$ ).
Compare $P(A \text{ and } B)$ with $P(A) \times P(B)$ : First, calculate the product of $P(A)$ and $P(B)$ . $P(A) \times P(B) = \left(\frac{1}{2}\right) \times \left(\frac{4}{7}\right) = \frac{4}{14} = \frac{2}{7}$ .
Conclusion: Now, compare $P(A \text{ and } B)$ with $P(A) \times P(B)$ . $\newline$ $P(A \text{ and } B) = \frac{1}{7}$ , but we calculated $P(A) \times P(B) = \frac{2}{7}$ .
Conclusion: Now, compare $P(A \text{ and } B)$ with $P(A) \times P(B)$ . $P(A \text{ and } B) = \frac{1}{7}$ , but we calculated $P(A) \times P(B) = \frac{2}{7}$ . Since $P(A \text{ and } B)$ is not equal to $P(A) \times P(B)$ , events $A$ and $B$ are not independent.

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