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In an experiment, the probability that event $A$ occurs is $\frac{2}{7}$ , the probability that event $B$ occurs is $\frac{5}{7}$ , and the probability that events $A$ and $B$ both occur is $\frac{10}{49}$ . $\newline$ 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{2}{7}

, the probability that event

B

occurs is

\frac{5}{7}

, and the probability that events

A

and

B

both occur is

\frac{10}{49}

.

\newline

Are

A

and

B

independent events?

\newline

Choices:

\newline

(A) yes

\newline

(B) no

Calculate Product: Calculate the product of the individual probabilities of $A$ and $B$ . $P(A) \times P(B) = \left(\frac{2}{7}\right) \times \left(\frac{5}{7}\right)$
Perform Multiplication: Perform the multiplication to find the product. $\newline$ $(\frac{2}{7}) \times (\frac{5}{7}) = \frac{10}{49}$
Compare Probabilities: Compare the product of $P(A)$ and $P(B)$ with the probability of $A$ and $B$ occurring together. $\newline$ Since $P(A \text{ and } B) = \frac{10}{49}$ and $P(A) \times P(B) = \frac{10}{49}$ , the probabilities are equal.
Conclude Independence: Conclude whether $A$ and $B$ are independent based on the comparison. Since $P(A \text{ and } B)$ is equal to $P(A) \times P(B)$ , events $A$ and $B$ are independent.

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