Carl Horowitz

**University of Michigan**

Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

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The first is the probability that they have an opinion and basically what this is the probability of a yes or the probability of a no. These are mutually exclusive events, there is no way for you to say yes and no so all we have to do is take the probability of a yes and add it to the probability of a no.

There are a total of a hundred people, of those 55 said yes, of those 20 said no, so the probability of saying yes or no we just add those two together and we end up with 75 over 100. So this is 55 over 100 plus 20 over 100 giving us a grand total of 75 over 100 or three-fourths.

The other one we're dealing with is the probability that we have a guy for undecided, so here is some overlap because we have guys who are undecided right here and one way we could do this is basically just add up the guys and undecided and go from there, but I want to use our formula so we can see how this works.

So what we are dealing with is the probability that we have a guy. There are 60 guys out of 100 so this is just going to be 60 out of 100 is a probability that we have a guy.

Probability that we are undecided, so when we have this overlap, we still add them together, probability of undecided is just going to be the 25, undecided out of a 100 in total, so plus 25 out of 100 and then we have to minus what we double counted. We counted the guys who are undecided twice, we counted them in both of the guy and undecided, so we have to subtract off the probability that we have a guy who is undecided no opinion, so that's going to be subtract 20 out of 100 which works our to be 60 plus 25 is 85 minus 20 65 out of 100 whatever that simplifies out to.

So what we did is two different ways of dealing with mutual events. First one is mutually exclusive, so we can just add the probabilities together, the second one there is some overlap so we have to add them together, but then subtract off that overlap that we double counted.

So dealing with the probability of multiple events, just going with our formulas one for no overlap, another with overlap.