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