Hey everyone. We just learned how to find the probability that some event will happen, like, say, rolling a 6-sided die and getting a 4. But what about the probability that some event will not happen? Well, this is something that you'll actually be asked explicitly to calculate and it may sound like it's going to be tricky, but you can actually do it using something that we already know. So here I'm going to walk you through how to calculate the probability that some event will not happen by simply using the probability that it will.
So let's go ahead and get started here. Now all of the possible outcomes where some event does not happen actually have its own special name, and it's referred to as the complement of that event. So when looking at my dice roll here, if I consider all of the possible outcomes of not rolling a 4, like rolling a 1, 2, 3, 5, or 6, all of these outcomes together represent the complement of rolling a 4. Now if we refer to our event as a, we can use a special notation to denote the complement of a. So you may see this written as
So in this case, there's only one way I could roll a 4 divided by the number of total possible outcomes. So since this is a 6 sided die, all of my total outcomes are 6. So the probability of
So here, I take all of the outcomes that include that event, not rolling a 4, and divide it by the number of total possible outcomes, in this case still 6. Now looking at these, if I were to take the probability of
The probability of some event plus the probability of its complement is going to be equal to 1. And we can use this formula over here to more easily calculate the probability of something not happening by rearranging a little bit here. So if I were to subtract the probability of
In this example, I'm asked when drawing a single card from a standard deck of 52, what is the probability that I will not draw a queen? Well, instead of trying to find all of the cards that are not a queen, let's just consider all of the cards that are queens. So if I look at the probability of getting a queen, I know that in a standard deck of 52 cards, there are 4 queens so I take all of the outcomes that include my event drawing a queen and put that over the number of total possible outcomes. In this case, since I have 52 total cards, my total is 52. Then to find the probability of not drawing a queen, I can simply take 1 minus the probability of drawing that queen, which we just calculated.
So we can go ahead and plug in that
Thanks for watching. And I'll see you in the next one.