In an experiment, the probability that event A occurs is 6/7, the probability that event B occurs is 8/9, and the probability that events A and B both occur is 16/21. 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, calculate the product of P(A) and P(B). P(A) * P(B) = (6/7) * (8/9)Perform Multiplication: Now, do the multiplication. (6/7) * (8/9) = 48/63Simplify Fraction: Simplify the fraction 48/63 to its lowest terms. 48/63 = 16/21Compare Products: Compare the simplified product to P(A and B). Since P(A and B) = 16/21 and P(A) * P(B) = 16/21, they are equal.Verify 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 $\frac{6}{7}$ , the probability that event $B$ occurs is $\frac{8}{9}$ , and the probability that events $A$ and $B$ both occur is $\frac{16}{21}$ . $\newline$ 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{6}{7}

, the probability that event

B

occurs is

\frac{8}{9}

, and the probability that events

A

and

B

both occur is

\frac{16}{21}

\newline

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, calculate the product of $P(A)$ and $P(B)$ . $\newline$ $P(A) \times P(B) = \left(\frac{6}{7}\right) \times \left(\frac{8}{9}\right)$
Perform Multiplication: Now, do the multiplication. $\newline$ $(\frac{6}{7}) \times (\frac{8}{9}) = \frac{48}{63}$
Simplify Fraction: Simplify the fraction $\frac{48}{63}$ to its lowest terms. $\newline$ $\frac{48}{63} = \frac{16}{21}$
Compare Products: Compare the simplified product to $P(A \text{ and } B)$ . $\newline$ Since $P(A \text{ and } B) = \frac{16}{21}$ and $P(A) \times P(B) = \frac{16}{21}$ , they are equal.
Verify Independence: Since $P(A \text{ and } B)$ is equal to $P(A) \times P(B)$ , events $A$ and $B$ are independent.