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In an experiment, the probability that event $A$ occurs is $\frac{6}{7}$ , the probability that event $B$ occurs is $\frac{3}{8}$ , and the probability that event $A$ occurs given that event $B$ occurs is $\frac{6}{7}$ . Are $A$ and $B$ independent events?

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Identify independent events

Full solution

Q. In an experiment, the probability that event

A

occurs is

\frac{6}{7}

, the probability that event

B

occurs is

\frac{3}{8}

, and the probability that event

A

occurs given that event

B

occurs is

\frac{6}{7}

. Are

A

and

B

independent events?

Given Information: We are given: $\newline$ $P(A) = \frac{6}{7}$ $\newline$ $P(B) = \frac{3}{8}$ $\newline$ $P(A|B) = \frac{6}{7}$ $\newline$ Identify the definition of independent events. Two events $A$ and $B$ are independent if the occurrence of one does not affect the probability of the occurrence of the other, which mathematically means $P(A|B) = P(A)$ .
Definition of Independent Events: Since we are given $P(A|B)$ , we can use the definition of conditional probability to find $P(A \cap B)$ . The formula for conditional probability is $P(A|B) = \frac{P(A \cap B)}{P(B)}$ .
Calculate $P(A \cap B)$ : We can rearrange the formula to find $P(A \cap B)$ :
$P(A \cap B) = P(A|B) \times P(B)$
Substitute the given probabilities into the equation:
$P(A \cap B) = (\frac{6}{7}) \times (\frac{3}{8})$
Check for Independence: Perform the multiplication to find $P(A \cap B)$ : $P(A \cap B) = \frac{6}{7} \times \frac{3}{8}$ $P(A \cap B) = \frac{18}{56}$ $P(A \cap B) = \frac{9}{28}$
Check for Independence: Perform the multiplication to find $P(A \cap B)$ : $P(A \cap B) = \frac{6}{7} \times \frac{3}{8}$ $P(A \cap B) = \frac{18}{56}$ $P(A \cap B) = \frac{9}{28}$ Now, let's check if $P(A \cap B)$ is equal to $P(A) \times P(B)$ to determine if events A and B are independent. $P(A) \times P(B) = \frac{6}{7} \times \frac{3}{8}$ $P(A) \times P(B) = \frac{18}{56}$ $P(A) \times P(B) = \frac{9}{28}$
Check for Independence: Perform the multiplication to find $P(A \cap B)$ : $P(A \cap B) = \frac{6}{7} \times \frac{3}{8}$ $P(A \cap B) = \frac{18}{56}$ $P(A \cap B) = \frac{9}{28}$ Now, let's check if $P(A \cap B)$ is equal to $P(A) \times P(B)$ to determine if events A and B are independent. $P(A) \times P(B) = \frac{6}{7} \times \frac{3}{8}$ $P(A) \times P(B) = \frac{18}{56}$ $P(A) \times P(B) = \frac{9}{28}$ Since $P(A \cap B) = P(A) \times P(B)$ , we conclude that events A and B are independent because the joint probability is equal to the product of the individual probabilities.

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