In an experiment, the probability that event A occurs is 4/9, the probability that event B occurs is 6/7, and the probability that events A and B both occur is 8/21. 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) = (4/9) * (6/7)Perform Multiplication: Now, do the multiplication. (4/9) * (6/7) = 24/63Simplify Fraction: Simplify the fraction 24/63. 24/63 can be simplified to 8/21 by dividing both the numerator and the denominator by 3.Compare Probabilities: Now, compare P(A and B) with the product P(A) * P(B). P(A and B) = 8/21 and P(A) * P(B) = 8/21Confirm Independence: Since P(A and B) is equal to P(A) * P(B), events A and B are independent.

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In an experiment, the probability that event $A$ occurs is $\frac{4}{9}$ , the probability that event $B$ occurs is $\frac{6}{7}$ , and the probability that events $A$ and $B$ both occur is $\frac{8}{21}$ . $\newline$ 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}{9}

, the probability that event

B

occurs is

\frac{6}{7}

, and the probability that events

A

and

B

both occur is

\frac{8}{21}

\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 $P(B)$ . $P(A) \times P(B) = \frac{4}{9} \times \frac{6}{7}$
Perform Multiplication: Now, do the multiplication. $\newline$ $(\frac{4}{9}) \times (\frac{6}{7}) = \frac{24}{63}$
Simplify Fraction: Simplify the fraction $\frac{24}{63}$ . $\frac{24}{63}$ can be simplified to $\frac{8}{21}$ by dividing both the numerator and the denominator by $3$ .
Compare Probabilities: Now, compare $P(A \text{ and } B)$ with the product $P(A) \times P(B)$ . $P(A \text{ and } B) = \frac{8}{21}$ and $P(A) \times P(B) = \frac{8}{21}$
Confirm Independence: Since $P(A \text{ and } B)$ is equal to $P(A) \times P(B)$ , events $A$ and $B$ are independent.