An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Calculate P ( Z ∣ B ) = 280 126 = 0.45 .
Calculate P ( Z ) = 660 297 = 0.45 .
Since P ( Z ∣ B ) = P ( Z ) , Z and B are independent.
Therefore, the correct statement is: Z and B are independent events because P ( Z mi d B ) = P ( Z ) .
Explanation
Analyze the problem We are given a contingency table and asked to determine if events Z and B are independent. Two events are independent if and only if the conditional probability of one event occurring given that the other event has occurred is equal to the probability of the first event occurring. In other words, Z and B are independent if P ( Z ∣ B ) = P ( Z ) . We will calculate these probabilities and compare them.
Calculate P(Z|B) First, we need to calculate P ( Z ∣ B ) , which is the probability of event Z occurring given that event B has occurred. This is calculated as: P ( Z ∣ B ) = P ( B ) P ( Z ∩ B ) From the table, we see that the number of times both Z and B occur is 126. The total number of times B occurs is 280. Therefore, P ( Z ∩ B ) = 660 126 P ( B ) = 660 280 So, P ( Z ∣ B ) = 660 280 660 126 = 280 126 = 0.45
Calculate P(Z) Next, we need to calculate P ( Z ) , which is the probability of event Z occurring. This is calculated as: P ( Z ) = Total number of observations Number of times Z occurs From the table, we see that the number of times Z occurs is 297, and the total number of observations is 660. Therefore, P ( Z ) = 660 297 = 0.45
Compare P(Z|B) and P(Z) Now we compare P ( Z ∣ B ) and P ( Z ) . We found that P ( Z ∣ B ) = 0.45 and P ( Z ) = 0.45 . Since P ( Z ∣ B ) = P ( Z ) , events Z and B are independent.
Conclusion Finally, we can conclude that Z and B are independent events because P ( Z ∣ B ) = P ( Z ) .
Examples
Contingency tables and independence of events are used in various fields such as marketing, healthcare, and social sciences. For example, a marketing analyst might use a contingency table to analyze the relationship between advertising campaigns (A, B, C) and customer demographics (X, Y, Z) to determine if a particular campaign is more effective for a specific demographic. If the analyst finds that a campaign and a demographic are independent, it means that the campaign's effectiveness is not influenced by that demographic, and the marketing strategy can be adjusted accordingly.
The number of electrons flowing through an electric device that delivers a current of 15.0 A for 30 seconds is approximately 2.81 billion billion electrons. This is calculated using the relationship between current, charge, and the fundamental charge of an electron. By calculating the total charge and dividing it by the charge of a single electron, we arrive at this large number.
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