In an experiment, the probability that event A occurs is 4/7, the probability that event B occurs is 1/2, and the probability that events A and B both occur is 1/7. 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, calculate the product of P(A) and P(B). P(A) * P(B) = (4/7) * (1/2)Compare Product to Joint Probability: Now, do the multiplication. (4/7) * (1/2) = 4/14 = 2/7Conclusion: Compare this product to the probability of A and B both occurring, which is given as 1/7. P(A and B) = 1/7Since 2/7 is not equal to 1/7, the product of P(A) and P(B) is not equal to P(A and B). Therefore, events A and B are not independent.

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In an experiment, the probability that event $A$ occurs is $\frac{4}{7}$ , the probability that event $B$ occurs is $\frac{1}{2}$ , and the probability that events $A$ and $B$ both occur is $\frac{1}{7}$ . $\newline$ Are $A$ and $B$ independent events? $\newline$ Choices: $\newline$ (A) yes $\newline$ (B) no

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Math Problems

Precalculus

Identify independent events

Full solution

Q. In an experiment, the probability that event

A

occurs is

\frac{4}{7}

, the probability that event

B

occurs is

\frac{1}{2}

, and the probability that events

A

and

B

both occur is

\frac{1}{7}

\newline

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, calculate the product of $P(A)$ and $P(B)$ . $\newline$ $P(A) \times P(B) = \left(\frac{4}{7}\right) \times \left(\frac{1}{2}\right)$
Compare Product to Joint Probability: Now, do the multiplication. $\newline$ $(\frac{4}{7}) \times (\frac{1}{2}) = \frac{4}{14} = \frac{2}{7}$
Conclusion: Compare this product to the probability of $A$ and $B$ both occurring, which is given as $\frac{1}{7}$ . $P(A \text{ and } B) = \frac{1}{7}$
Conclusion: Compare this product to the probability of $A$ and $B$ both occurring, which is given as $\frac{1}{7}$ . $P(A \text{ and } B) = \frac{1}{7}$ Since $\frac{2}{7}$ is not equal to $\frac{1}{7}$ , the product of $P(A)$ and $P(B)$ is not equal to $P(A \text{ and } B)$ . Therefore, events $A$ and $B$ are not independent.