Determining If Two Events Are Independent or Dependent. The following probabilities for choosing 2 marbles from a bag of 10 marbles, determine if the events are independent or independent. [tex]P(blue)=\frac{3}{10}, P(green)=\frac{1}{5}, \text{ and } P(blue and green)=\frac{3}{100}[/tex] A. Independent, because the product of [tex]P(blue)[/tex] and [tex]P(green)[/tex] does not equal [tex]P(blue and green)[/tex] B. Independent, because the product of [tex]P(blue)[/tex] and [tex]P(green)[/tex] equals [tex]P(blue and green)[/tex] C. Dependent, because the product of [tex]P(blue)[/tex] and [tex]P(green)[/tex] does not equal [tex]P(blue and green)[/tex] D. Dependent, because the product of [tex]P(blue)[/tex] and [tex]P(green)[/tex] equals [tex]P(blue and green)[/tex]
Answer (2)
Calculate the product of the probabilities of the individual events: P ( blue ) × P ( green ) = 10 3 × 5 1 = 50 3 .
Compare the result with the probability of both events occurring: P ( blue and green ) = 100 3 .
Since 50 3 = 100 3 , the events are dependent.
The events are dependent because P ( blue ) × P ( green ) = P ( blue and green ) . Therefore, the answer is: $\boxed{\text{Dependent}}.
Explanation
Understand the problem and provided data We are given the probabilities of two events, 'blue' and 'green', and the probability of both events occurring. We need to determine whether these events are independent or dependent. Two events are independent if the probability of both occurring is equal to the product of their individual probabilities.
List given probabilities We have the following probabilities:
P ( blue ) = 10 3
P ( green ) = 5 1
P ( blue and green ) = 100 3
Calculate the product of individual probabilities To check for independence, we need to calculate the product of P ( blue ) and P ( green ) and compare it to P ( blue and green ) .
P ( blue ) × P ( green ) = 10 3 × 5 1 = 50 3
Compare the calculated product with the given probability Now we compare the calculated product with the given probability of both events occurring:
50 3 and 100 3
Since 50 3 = 100 3 , the events are not independent. They are dependent.
State the final answer Since the product of P ( blue ) and P ( green ) is not equal to P ( blue and green ) , the events 'blue' and 'green' are dependent.
So, the correct answer is: Dependent, because the product of P ( blue ) and P ( green ) does not equal $P(\text{blue and green}).
Examples
In weather forecasting, determining if rainfall and cloudy skies are independent events can help improve predictions. If the probability of rain and clouds occurring together is different from the product of their individual probabilities, it suggests that other factors are influencing the weather patterns.
The events of choosing a blue marble and a green marble are dependent because the product of their individual probabilities does not equal the probability of both occurring together. The calculated product is \frac{6}{100}, which is different from \frac{3}{100}. Therefore, the correct answer is option C.
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