Using the Complement to Calculate Probability - Problem 1
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 and typically the key word you're going to see in these instances is at least, so the at least is telling us that we're dealing with a number of different situations.
Behind me I have a pretty straight forward example, so we have a chart that says the number of hour people spend watching TV and the number of kids who spend each time period. We're looking for the probability that you watch at least one hour of TV of someone just watching at random.
So if you say at least what that would have to be is the probability that they watch 1 to 2, plus the probability to watch 2 to 3, plus the probability to 3 to 4, and 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 one 1 to 2 is just going to be 30 out of 105 add up to 2 to 3 that was going to be 10 out of 105.
So we could add up these all together fairly easily, but this is an example to show you how the complement works. So what we can do is we know that if we add up all of these not just with one hour we add 0 all the way to 5 plus, we're going to have a probability of 1, choose everything in an event, the probability has to be 1.
So what we can do is then this probability of watching at least one hour is going to be the same thing as 1 minus the probability of not 1 plus hours. So this is the complement now watching more than an hour and this one is really easy to follow because now 1 plus hours is just same watching to 0 to 1, so 20 students watch 0 to 1 hours and so all we have to do is just calculate 1 minus 20 over 105 which is going to give us 85 a hundred and fifths. We could add up these four categories, we should get the same thing so 30 plus 10 is 40 plus 40 is 80 plus 5 is 85.
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.
So we have a couple of general formulas for using this and basically the main thing is that the probability of an event plus the probability of its complement written E this little c is going to be 1, probability of everything has to be 1 and then basically you can subtract over E to have these to the other side and so you get the probability of an event is equal to 1 minus the probability of a complement. And like I said this example is pretty straight forward, but when we get to more complicated probabilities, this can come in really handy.