In an experiment, the probability that event A occurs is 2/7, the probability that event B occurs is 7/9, and the probability that events A and B both occur is 1/9. Are A and B independent events? Choices: [[yes][no]]

Define Independent Events: Define the concept of independent events. Two events, A and B, are independent if the occurrence of one does not affect the probability of the occurrence of the other. Mathematically, this means that if A and B are independent, then 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 Individual Probabilities: Calculate the product of the individual probabilities of events A and B. P(A) * P(B) = (2/7) * (7/9) To calculate this product, we multiply the numerators together and the denominators together. (2 * 7) / (7 * 9) = 14 / 63 We can simplify this fraction by dividing both the numerator and the denominator by 7. 14 / 63 = 2 / 9Compare Probabilities: Compare the product of the individual probabilities to the probability of both events occurring together. We have calculated that P(A) * P(B) = 2/9. Now we need to compare this to P(A and B), which is given as 1/9. If P(A) * P(B) = P(A and B), then events A and B are independent. If not, they are dependent. P(A) * P(B) = 2/9 P(A and B) = 1/9 Since 2/9 is not equal to 1/9, the product of the individual probabilities does not equal the probability of both events occurring together.Conclude Independence: Conclude whether events A and B are independent or dependent. Since P(A) * P(B) ≠ P(A and B), events A and B are not independent. Therefore, they are dependent events.

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In an experiment, the probability that event $A$ occurs is $\frac{2}{7}$ , the probability that event $B$ occurs is $\frac{7}{9}$ , and the probability that events $A$ and $B$ both occur is $\frac{1}{9}$ . $\newline$ Are $A$ and $B$ independent events? $\newline$ Choices: $\newline$ (A) $\text{yes}$ $\newline$ (B) $\text{no}$

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Math Problems

Algebra 2

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{7}{9}

, and the probability that events

A

and

B

both occur is

\frac{1}{9}

\newline

Are

A

and

B

independent events?

\newline

Choices:

\newline

(A)

\text{yes}

\newline

(B)

\text{no}

Define Independent Events: Define the concept of independent events. Two events, $A$ and $B$ , are independent if the occurrence of one does not affect the probability of the occurrence of the other. Mathematically, this means that if $A$ and $B$ are independent, then 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 Individual Probabilities: Calculate the product of the individual probabilities of events $A$ and $B$ . $P(A) \times P(B) = \left(\frac{2}{7}\right) \times \left(\frac{7}{9}\right)$ To calculate this product, we multiply the numerators together and the denominators together. $\left(2 \times 7\right) / \left(7 \times 9\right) = \frac{14}{63}$ We can simplify this fraction by dividing both the numerator and the denominator by $7$ . $\frac{14}{63} = \frac{2}{9}$
Compare Probabilities: Compare the product of the individual probabilities to the probability of both events occurring together. $\newline$ We have calculated that $P(A) \times P(B) = \frac{2}{9}$ . Now we need to compare this to $P(A \text{ and } B)$ , which is given as $\frac{1}{9}$ . $\newline$ If $P(A) \times P(B) = P(A \text{ and } B)$ , then events A and B are independent. If not, they are dependent. $\newline$ $P(A) \times P(B) = \frac{2}{9}$ $\newline$ $P(A \text{ and } B) = \frac{1}{9}$ $\newline$ Since $\frac{2}{9}$ is not equal to $\frac{1}{9}$ , the product of the individual probabilities does not equal the probability of both events occurring together.
Conclude Independence: Conclude whether events $A$ and $B$ are independent or dependent. Since $P(A) \times P(B) \neq P(A \text{ and } B)$ , events $A$ and $B$ are not independent. Therefore, they are dependent events.