In an experiment, the probability that event A occurs is 3/8, the probability that event B occurs is 5/6, and the probability that events A and B both occur is 5/16. Are A and B independent events? Choices: (A)yes (B)no

Calculate Product: 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)).Multiply Probabilities: First, calculate the product of P(A) and P(B). P(A) * P(B) = (3/8) * (5/6).Simplify Fraction: Now, do the multiplication. (3/8) * (5/6) = 15/48.Compare to P(A and B): Simplify the fraction 15/48 to its lowest terms. 15/48 = 5/16.Verify Independence: Compare the simplified product to P(A and B). P(A and B) = 5/16, and P(A) * P(B) = 5/16.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 $\frac{3}{8}$ , the probability that event $B$ occurs is $\frac{5}{6}$ , and the probability that events $A$ and $B$ both occur is $\frac{5}{16}$ . $\newline$ 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{3}{8}

, the probability that event

B

occurs is

\frac{5}{6}

, and the probability that events

A

and

B

both occur is

\frac{5}{16}

\newline

Are

A

and

B

independent events?

\newline

Choices:

\newline

(A) yes

\newline

(B) no

Calculate Product: 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 ( $P(A) \times P(B)$ ).
Multiply Probabilities: First, calculate the product of $P(A)$ and $P(B)$ . $\newline$ $P(A) \times P(B) = \left(\frac{3}{8}\right) \times \left(\frac{5}{6}\right).$
Simplify Fraction: Now, do the multiplication. $\newline$ $(\frac{3}{8}) \times (\frac{5}{6}) = \frac{15}{48}$ .
Compare to $P(A \text{ and } B)$ : Simplify the fraction $\frac{15}{48}$ to its lowest terms. $\frac{15}{48} = \frac{5}{16}$ .
Verify Independence: Compare the simplified product to $P(A \text{ and } B)$ . $P(A \text{ and } B) = \frac{5}{16}$ , and $P(A) \times P(B) = \frac{5}{16}$ .
Verify Independence: Compare the simplified product to $P(A \text{ and } B)$ . $P(A \text{ and } B) = \frac{5}{16}$ , and $P(A) \times P(B) = \frac{5}{16}$ .Since $P(A \text{ and } B)$ is equal to $P(A) \times P(B)$ , events A and B are independent.