In an experiment, the probability that event A occurs is 2/5, the probability that event B occurs is 7/8, and the probability that events A and B both occur is 7/20. 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) = (2/5) * (7/8)Simplify Fraction: Now, do the multiplication. (2/5) * (7/8) = 14/40 But we need to simplify this fraction.Compare P(A and B): Simplify 14/40 to its lowest terms. 14/40 = 7/20Events A and B are Independent: Now, compare P(A and B) with the product of P(A) * P(B). P(A and B) = 7/20 P(A) * P(B) = 7/20Since 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 $\frac{2}{5}$ , the probability that event $B$ occurs is $\frac{7}{8}$ , and the probability that events $A$ and $B$ both occur is $\frac{7}{20}$ . $\newline$ 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}{5}

, the probability that event

B

occurs is

\frac{7}{8}

, and the probability that events

A

and

B

both occur is

\frac{7}{20}

\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 $P(B)$ . $\newline$ $P(A) \times P(B) = \left(\frac{2}{5}\right) \times \left(\frac{7}{8}\right)$
Simplify Fraction: Now, do the multiplication. $\newline$ $(\frac{2}{5}) \times (\frac{7}{8}) = \frac{14}{40}$ $\newline$ But we need to simplify this fraction.
Compare $P(A \text{ and } B)$ : Simplify $\frac{14}{40}$ to its lowest terms. $\newline$ $\frac{14}{40} = \frac{7}{20}$
Events A and B are Independent: Now, compare $P(A \text{ and } B)$ with the product of $P(A) \times P(B)$ . $\newline$ $P(A \text{ and } B) = \frac{7}{20}$ $\newline$ $P(A) \times P(B) = \frac{7}{20}$
Events A and B are Independent: Now, compare $P(A \text{ and } B)$ with the product of $P(A) \times P(B)$ . $\newline$ $P(A \text{ and } B) = \frac{7}{20}$ $\newline$ $P(A) \times P(B) = \frac{7}{20}$ Since $P(A \text{ and } B)$ is equal to $P(A) \times P(B)$ , events A and B are independent.