In an experiment, the probability that event A occurs is 2/3, the probability that event B occurs is 1/3, and the probability that events A and B both occur is 1/9. Are A and B independent events? Choices: (A)yes (B)no

Check Independence Criteria: To check if events A and B are independent, we need to see if the probability of A and B occurring together (P(A and B)) is equal to the product of their individual probabilities (P(A) * P(B)).Calculate Product of Probabilities: First, let's find the product of P(A) and P(B). P(A) * P(B) = (2/3) * (1/3) = 2/9.Compare Product to Joint Probability: Now, let's compare this product to the probability of A and B occurring together, which is given as 1/9.Conclusion of Independence: Since 2/9 is not equal to 1/9, the probability of A and B occurring together is not the same as the product of their individual probabilities.Therefore, events A and B are not independent.

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In an experiment, the probability that event $A$ occurs is $\frac{2}{3}$ , the probability that event $B$ occurs is $\frac{1}{3}$ , and the probability that events $A$ and $B$ both occur is $\frac{1}{9}$ . Are $A$ and $B$ independent events? $\newline$ Choices: $\newline$ (A)yes $\newline$ (B)no

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Math Problems

Precalculus

Identify independent events

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Q. In an experiment, the probability that event

A

occurs is

\frac{2}{3}

, the probability that event

B

occurs is

\frac{1}{3}

, and the probability that events

A

and

B

both occur is

\frac{1}{9}

. Are

A

and

B

independent events?

\newline

Choices:

\newline

(A)yes

\newline

(B)no

Check Independence Criteria: 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 ( $P(A) \times P(B)$ ).
Calculate Product of Probabilities: First, let's find the product of $P(A)$ and $P(B)$ . $P(A) \times P(B) = \left(\frac{2}{3}\right) \times \left(\frac{1}{3}\right) = \frac{2}{9}$ .
Compare Product to Joint Probability: Now, let's compare this product to the probability of A and B occurring together, which is given as $\frac{1}{9}$ .
Conclusion of Independence: Since $\frac{2}{9}$ is not equal to $\frac{1}{9}$ , the probability of $A$ and $B$ occurring together is not the same as the product of their individual probabilities.
Conclusion of Independence: Since $\frac{2}{9}$ is not equal to $\frac{1}{9}$ , the probability of $A$ and $B$ occurring together is not the same as the product of their individual probabilities.Therefore, events $A$ and $B$ are not independent.