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Combinations - Problem 3
So next problem we're going to look at combined choosing with probability and what we're dealing with is a recent school election where 40 students voted for candidate A, 60 voted for candidate B and if 8 students are chosen at random, find the probability that 5 voted for candidate B.
So we know what we're talking about probability and whenever we deal with probability, what we're concerned with is the events that, the number of events that we want over the total number of outcomes. And so our total number of outcomes involve these 8 students that we chose. So we chose 8 students out of the entire class, there are 40 and 60 voting for each, so that tells us 100 students so that means our total number of outcomes are 100 choose 8.
The next thing we need to do is figure out the outcomes we want and so that is going to involve 5 students that voted for B and the other 3 that voted for candidate A. So of the 5 that voted for B, what we're concerned with is that we started with 60 of them, so the probability of 5 voting out of the 60 is 60 choose 5, so that's out of the 8 that we chose, 5 of which voted out of these 60 for candidate B.
Doing a similar thing for the other candidate, we're now concerned with 3 other candidates that came out of this group of 40, so that's just going to be 40 choose 3. So what we end up with is this fraction, 100 choose 8 is the total number of possible outcomes, 60 choose 5 is out of these 60 students we chose 5 of them out of the 8 to vote for them and out of the 40, we are choosing 3 to vote for candidate A, so by going through a combination of choosing in probability we were able to solve this problem.