In an experiment, the probability that event A occurs is 2/7, the probability that event B occurs is 5/6, and the probability that events A and B both occur is 5/21. 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) = (2/7) * (5/6)Multiply Probabilities: Now, do the multiplication. (2/7) * (5/6) = 10/42Simplify Fraction: Simplify the fraction 10/42 to its lowest terms. 10/42 = 5/21Compare P(A and B) with Product: Now, compare P(A and B) with the product of P(A) * P(B). P(A and B) = 5/21 and P(A) * P(B) = 5/21Confirm 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 $\frac{2}{7}$ , the probability that event $B$ occurs is $\frac{5}{6}$ , and the probability that events $A$ and $B$ both occur is $\frac{5}{21}$ . $\newline$ 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{2}{7}

, the probability that event

B

occurs is

\frac{5}{6}

, and the probability that events

A

and

B

both occur is

\frac{5}{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 $P(B)$ . $P(A) \times P(B) = \frac{2}{7} \times \frac{5}{6}$
Multiply Probabilities: Now, do the multiplication. $\newline$ $(\frac{2}{7}) \times (\frac{5}{6}) = \frac{10}{42}$
Simplify Fraction: Simplify the fraction $\frac{10}{42}$ to its lowest terms. $\newline$ $\frac{10}{42} = \frac{5}{21}$
Compare $P(A \text{ and } B)$ with Product: Now, compare $P(A \text{ and } B)$ with the product of $P(A) \times P(B)$ . $P(A \text{ and } B) = \frac{5}{21}$ and $P(A) \times P(B) = \frac{5}{21}$
Confirm Independence: Since $P(A \text{ and } B)$ is equal to $P(A) \times P(B)$ , events $A$ and $B$ are independent.