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 \(a'\), \(\overline{a}\), or \(\lnot a\). All of these are different ways to denote the complement of \(\mathit{a}\). Now that we know what the complement is, let's dive deeper into our dice roll example here. So in this example, we're asked, when rolling a 6-sided die, what is the probability that we will roll a 4? Now if we refer to this event as a, the probability of \(a\) is equal to the number of outcomes that include that event.
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 \(\mathit{a}\) is one-sixth. But what about the probability that we will not roll a 4 or the probability of the complement of \(\mathit{a}\)? Well, I know that I have 5 possible outcomes that I would not roll a 4 because I could roll a 1, 2, 3, 5, or 6.
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 \(\mathit{a}\) and the probability of its complement and add them together, I see that I get 6 over 6, which is just 1. Now, this makes sense, right, because we've covered 100% of the possibilities of rolling a 6-sided die, rolling a 1, 2, 3, 5, 6, or a 4. So it makes sense that the total probability of all possible events is simply 1. Now, this is always going to be true.
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 \(\mathit{a}\) from both sides here, it will cancel on that left side, leaving me to see that the probability of the complement of \(\mathit{a}\) is equal to 1 minus the probability of \(\mathit{a}\). So here, I see that the probability that something does not happen is simply 1 minus the probability that it will happen. So now that we know this formula, let's apply it to another example here.
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 \(\frac{4}{52}\) here. Now I know that one is simply the same thing as \(\frac{52}{52}\), just getting a common denominator here. So if I perform this subtraction, I end up with \(\frac{48}{52}\), which as a decimal is 0.92. So the probability of not drawing a queen is 0.92, and we found that by simply using the probability of drawing a queen without having to count up those 48 cards. So now that we know how to find the probability of something not happening, let's get some more practice.
Thanks for watching. And I'll see you in the next one.
