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Using the Complement to Calculate Probability - Concept

Teacher/Instructor Carl Horowitz
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!

Sometimes calculating probability can be fairly complicated, but we have tricks to make it easier. Calculating probability of the complement of an event can be easier than calculating the probability of the event itself. We can use the probability of the complement to find the probability of the event by subtracting it from one. This trick is often used when calculating the probability of multiple events.

Sometimes it's pretty hard to find the probability of an event occurring and so what we do is we actually use the complement of the event to find the probability we're looking for. So what the complement is, is basically the probability of something not happening, okay and typically the key word you're going to see in these instances is at least. So at least is telling us that we're dealing with a number of different situations. Behind me I've a pretty straight forward example okay, so we have a chart that says that the number of hours people spend watching TV and the number of kids who spend each time period.
Okay, we're working for the probability that you watch at least 1 hour of TV of someone just chosen out random. Okay so if we're see at least what that would have to be is the probability that they watch 1 to 2 plus the probability watch 2 to 3 plus the probability your 3 to 4 so on and so forth. Each of these isn't that hard in it's own right, we have 105 students total so basically probability they watch 1 to 2 is just going to be 30 out of 105. Okay add that to 2 to 3 that was going to be 10 out of 105.
Okay, so we can add up these all together fairly easily but this an example to show you just sort of how the complement works. So what we can do is we know that if we add up all of these not just with 1 hour we add 0 all the way to 5 plus we're going to have a probability of 1. Everything you do choose everything in an event the probability has to be 1 so what we can do is then this probability that watch at least 1 hour is going to be the same thing as 1 minus the probability of not 1 plus hours. So this is the complement, not watching more than an hour. And this one is really easy to follow because not 1 plus hours is just the same as watching to 0 to 1.
Okay, so 20 students watch 0 to 1 hours and so all we have to do is just calculate 1 minus 20 over 105 is just going to gives us 85 a hundred and fifths. Okay we could add up these 4 categories we should get the same thing, so 30 plus 10 is 40 plus 40 is 80 plus 5 is 85 okay so using the fact that the probability of all events add up to 1 we can easily use the complement when things can get a little bit more complicated. Okay so we have a couple of general formulas for using this, and basically the main thing is is that the probability of an event plus the probability of its complement written e with this little c is going to be 1. Probability of everything has to be 1 and then basically you can subtract over either of these to the other side and so you get the probability of the event is equal to 1 minus the probability of a complement.
Okay and like I said this example is pretty straight forward but when we get to more complicated probabilities this can come in really handy.

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