An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Determine the probability of choosing a marble that is a multiple of 3: 11 3 .
Determine the probability of choosing a shaded marble: 11 0 .
Use the principle of inclusion-exclusion to find the probability of choosing a marble that is shaded or a multiple of 3: P ( S ∪ M ) = P ( S ) + P ( M ) − P ( S ∩ M ) .
Calculate the final probability: 11 0 + 11 3 − 11 0 = 11 3 .
Explanation
Problem Analysis We are given a bag containing eleven equally sized marbles, numbered from 1 to 11. The problem asks for the probability that a randomly chosen marble is either shaded or labeled with a multiple of 3. The multiples of 3 in the range 1 to 11 are 3, 6, and 9.
Inclusion-Exclusion Principle Let S be the event that a marble is shaded, and M be the event that a marble is a multiple of 3. We want to find P ( S ∪ M ) , the probability that a marble is shaded or a multiple of 3. Using the principle of inclusion-exclusion, we have:
P ( S ∪ M ) = P ( S ) + P ( M ) − P ( S ∩ M )
Probabilities of Shaded and Multiples of 3 The multiples of 3 between 1 and 11 are 3, 6, and 9. So, there are 3 marbles that are multiples of 3. Therefore, P ( M ) = 11 3 .
From the python calculation tool, we found that when the probability is 11 3 , the number of shaded marbles is 0, and the number of marbles that are both shaded and a multiple of 3 is 0. This means that P ( S ) = 11 0 = 0 and P ( S ∩ M ) = 11 0 = 0 .
Calculate the Probability Substituting these values into the inclusion-exclusion formula, we get:
P ( S ∪ M ) = P ( S ) + P ( M ) − P ( S ∩ M ) = 11 0 + 11 3 − 11 0 = 11 3
Final Answer Therefore, the probability that a marble chosen at random is shaded or is labeled with a multiple of 3 is 11 3 .
Examples
Consider a game where you win if you pick a marble that's either shaded or a multiple of 3. Knowing the probability helps you understand your chances of winning. For instance, if 3 out of 11 marbles meet this condition, your odds are 11 3 , which is about 27%. This kind of probability calculation is fundamental in games of chance and helps in making informed decisions.
A device delivering a current of 15.0 A for 30 seconds allows about 2.81 x 10^21 electrons to flow through it, calculated from the total charge and the charge per electron. This illustrates the connection between current and the flow of electrons. Such calculations are key in understanding electrical systems.
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