In an experiment, the probability that event A occurs is 2/7, the probability that event B occurs is 2/7, and the probability that events A and B both occur is 4/49. Are A and B independent events? Choices: (A)yes (B)no

Check for Independence: 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) = (2/7) * (2/7) = 4/49.Compare Product to Joint Probability: Now, let's compare this product to the probability of A and B occurring together, which is given as 4/49.Confirm Independence: Since P(A and B) = 4/49 and P(A) * P(B) = 4/49, the two probabilities are equal.Because the product of the individual probabilities is equal to the probability of both events occurring together, events A and B are independent.

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In an experiment, the probability that event $A$ occurs is $\frac{2}{7}$ , the probability that event $B$ occurs is $\frac{2}{7}$ , and the probability that events $A$ and $B$ both occur is $\frac{4}{49}$ . 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}{7}

, the probability that event

B

occurs is

\frac{2}{7}

, and the probability that events

A

and

B

both occur is

\frac{4}{49}

. Are

A

and

B

independent events?

\newline

Choices:

\newline

(A) yes

\newline

(B) no

Check for Independence: 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 $P(B)$ . $\newline$ $P(A) \times P(B) = \left(\frac{2}{7}\right) \times \left(\frac{2}{7}\right) = \frac{4}{49}.$
Compare Product to Joint Probability: Now, let's compare this product to the probability of $A$ and $B$ occurring together, which is given as $\frac{4}{49}$ .
Confirm Independence: Since $P(A \text{ and } B) = \frac{4}{49}$ and $P(A) \times P(B) = \frac{4}{49}$ , the two probabilities are equal.
Confirm Independence: Since $P(A \text{ and } B) = \frac{4}{49}$ and $P(A) \times P(B) = \frac{4}{49}$ , the two probabilities are equal. Because the product of the individual probabilities is equal to the probability of both events occurring together, events $A$ and $B$ are independent.