An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
Calculate the product of the probabilities of red and green: P ( re d ) × P ( g ree n ) = 2 1 × 4 1 = 8 1 .
Compare the result with the probability of both red and green: P ( re d ∩ g ree n ) = 8 1 .
Since P ( re d ) × P ( g ree n ) = P ( re d ∩ g ree n ) , the events are independent.
The correct statement is: The events are independent because P ( re d ) × P ( g ree n ) = P ( re d ∩ g ree n ) .
T h e e v e n t s a re in d e p e n d e n t b ec a u se P ( re d ) ⋅ P ( g ree n ) = P ( re d an d g ree n )
Explanation
State the given probabilities We are given the following probabilities:
P ( re d ) = 2 1
P ( g ree n ) = 4 1
P ( re d ∩ g ree n ) = 8 1
Condition for independence To determine if the events 'red' and 'green' are independent, we need to check if the following condition holds:
P ( re d ∩ g ree n ) = P ( re d ) ⋅ P ( g ree n )
Calculate the product of probabilities Let's calculate P ( re d ) ⋅ P ( g ree n ) :
P ( re d ) ⋅ P ( g ree n ) = 2 1 ⋅ 4 1 = 8 1
Compare the values Now, let's compare P ( re d ∩ g ree n ) with P ( re d ) ⋅ P ( g ree n ) :
We have P ( re d ∩ g ree n ) = 8 1 and P ( re d ) ⋅ P ( g ree n ) = 8 1 . Since these two values are equal, the events 'red' and 'green' are independent.
Calculate the sum of probabilities Now, let's consider the sum of the probabilities:
P ( re d ) + P ( g ree n ) = 2 1 + 4 1 = 4 2 + 4 1 = 4 3
Since P ( re d ∩ g ree n ) = 8 1 and P ( re d ) + P ( g ree n ) = 4 3 , we can see that P ( re d ∩ g ree n ) = P ( re d ) + P ( g ree n ) .
Conclusion Based on our calculations, the events are independent because P ( re d ) ⋅ P ( g ree n ) = P ( re d ∩ g ree n ) .
Examples
Understanding independent events is crucial in many real-world scenarios. For example, consider a marketing campaign where the probability of a customer clicking on an ad is independent of whether they saw a previous ad. If the probability of clicking on the first ad is 2 1 and the probability of clicking on the second ad is 4 1 , and the probability of clicking on both ads is 8 1 , then the events are independent. This knowledge helps marketers optimize their campaigns by understanding how different ads perform independently.
