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Problem 107 Hard Difficulty

Use the information in the Table 3.19 to answer the next eight exercises. The table shows the political party affiliation of each of 67 members of the US Senate in June 2012, and when they are up for reelection.
TABLE CANNOT COPY

The events “Republican” and “Up for reelection in 2016” are ________
a. mutually exclusive.
b. independent.
c. both mutually exclusive and independent.
d. neither mutually exclusive nor independent.

Answer

D "neither mutually exclusive nor independent:

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Video Transcript

Hi. We're looking at the two events Republican and up for reelection in 2016. And the question asked, It's multiple choice question. Are the mutually exclusive independent both or neither. And so let's take those two things one at a time. In terms of mutually exclusive, the question is half asking. Can they happen at the same time? So if we can calculate the probability, the probability that, um, it's Republican and re elected in November of 2016 then we know that it's not mutually exclusive. And so the probability of Republican and November 2016 would be 24 out of the 67 and we don't even necessarily need to calculate the decimal for that as long as we know that they both can happen at the same time. Because there's 24 senators that we're both Republican and re elected November 16 we know that it is not mutually exclusive, so no to this one, and then we need to check independent. Does one outcome affect the other? And so independence goes back to our formula if p p. So finding the probability of a given be that if that is equal to the probability of A than that shows us independence. And remember, the probability of a given B is equal to the probability of A and B over the probability of B. So what we're looking at for this question is to determine if the probability of Republican is equal to the probability of Republican and being re elected in November 2016 over the probability of the November 2016. So I just substituted. Since the probability of a given B is in both spots, I just set these two equal to each other. And then I substituted in the things that we were actually looking for. So we can find the probability of a Republican because there's 37 Republicans out of 67 senators, which gives us a probability of 0.55 to 2 for round figures. And then the probability of a Republican end being re elected November 2016 is 24 and they probably have a re election. November 2016 was 34 or the number of re elections, I should say, And when you divide those, we get 0.70 59 for round figures and those air not equal, so it is not independent. So in this case, your final answer would be D. It's neither mutually exclusive nor independent