A box contains different-colored marbles. If [tex]P(blue) = \frac{1}{4}[/tex], [tex]P(green) = \frac{1}{4}[/tex], and [tex]P(blue and green) = \frac{1}{12}[/tex], which statement is true?
A. The events are independent because [tex]P(blue) \cdot P(green) = P(blue and green)[/tex].
B. The events are independent because [tex]P(blue) \cdot P(green) \neq P(blue and green)[/tex].
C. The events are dependent because [tex]P(blue) \cdot P(green) = P(blue and green)[/tex].
D. The events are dependent because [tex]P(blue) \cdot P(green) \neq P(blue and green)[/tex].
Answer (1)
Calculate the product of the probabilities of the individual events: P ( b l u e ) ⋅ P ( g ree n ) = 4 1 ⋅ 4 1 = 16 1 .
Compare the result with the probability of both events occurring: P ( b l u e ∩ g ree n ) = 12 1 .
Since 16 1 = 12 1 , the events are dependent.
The events are dependent because P ( b l u e ) ⋅ P ( g ree n ) = P ( b l u e ∩ g ree n ) .
Explanation
Analyze the problem 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 the product of their individual probabilities. Otherwise, they are dependent.
Calculate the product of individual probabilities First, let's calculate the product of the individual probabilities of 'blue' and 'green': P ( b l u e ) ⋅ P ( g ree n ) = 4 1 ⋅ 4 1 = 16 1
Compare and conclude Now, let's compare this result with the given probability of both events occurring: P ( b l u e ∩ g ree n ) = 12 1 Since 16 1 = 12 1 , the events are not independent. They are dependent.
State the final answer Since P ( b l u e ) ⋅ P ( g ree n ) = P ( b l u e ∩ g ree n ) , the events 'blue' and 'green' are dependent. Therefore, the correct statement is: The events are dependent because P ( b l u e ) ⋅ P ( g ree n ) = P ( b l u e ∩ g ree n ) .
Examples
In weather forecasting, knowing if two events like 'rain' and 'thunder' are independent or dependent helps predict the likelihood of both happening together. If the probability of rain and thunder occurring together is different from the product of their individual probabilities, it indicates a dependency, possibly due to shared atmospheric conditions.
