In an experiment, the probability that event A occurs is 1/5, the probability that event B occurs is 7/9, and the probability that events A and B both occur is 7/45. 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) = (1/5) * (7/9)Multiply Probabilities: Now, do the multiplication. (1/5) * (7/9) = 7/45Compare Results: Next, compare this result to the given probability of A and B occurring together, which is 7/45.Confirm Independence: Since 7/45 equals 7/45, the product of P(A) and P(B) is equal to P(A and B).Conclusion: Therefore, events A and B are independent because the multiplication of their individual probabilities equals the probability of them occurring together.

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In an experiment, the probability that event $A$ occurs is $\frac{1}{5}$ , the probability that event $B$ occurs is $\frac{7}{9}$ , and the probability that events $A$ and $B$ both occur is $\frac{7}{45}$ . $\newline$ 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{1}{5}

, the probability that event

B

occurs is

\frac{7}{9}

, and the probability that events

A

and

B

both occur is

\frac{7}{45}

\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 $P(B)$ . $\newline$ $P(A) \times P(B) = \frac{1}{5} \times \frac{7}{9}$
Multiply Probabilities: Now, do the multiplication. $\newline$ $(\frac{1}{5}) \times (\frac{7}{9}) = \frac{7}{45}$
Compare Results: Next, compare this result to the given probability of $A$ and $B$ occurring together, which is $\frac{7}{45}$ .
Confirm Independence: Since $\frac{7}{45}$ equals $\frac{7}{45}$ , the product of $P(A)$ and $P(B)$ is equal to $P(A \text{ and } B)$ .
Conclusion: Therefore, events $A$ and $B$ are independent because the multiplication of their individual probabilities equals the probability of them occurring together.