 Hello, in this video we will talk about complementary events, basically events that are opposites of one another. The complement of an event A is denoted by A prime or A bar, once again that's A prime or A bar, that's the notation for the complement of an event A. It consists of all outcomes that are in the sample space but are not in the given event A. An event and it's complement cannot occur at the same time, that's the golden rule. A six-sided die is rolled. When I roll a six-sided die what would my sample space be? What are all of my possible outcomes? Well, anything from one to six, for a six-sided die one through six. So what is the probability of rolling a three? So the probability of rolling a three is equal to how many possible outcomes are there? How many items are there in my sample space? Six. Out of those six, how many are three? Just one. So the probability is one out of six. Now what is the probability of not rolling a three? Well the probability of not rolling a three is out of the six outcomes how many are not three? There's five of them. One, two, four, five, and six. Five outcomes. So the probability of not rolling a three is five out of six. So one thing I want to note before moving to the next slide is that notice the probability of rolling a three plus the probability of not rolling a three is equal to one. So notice that the probability of rolling a three plus not rolling a three is equal to one. If I was to rearrange this a little bit, remember that was one out of six plus five out of six is equal to one. If I was to rearrange this expression and isolate the probability of not rolling a three, I would take away the probability of rolling a three on both sides of the expression or the equation I should say. Notice what's going to happen. You're left with the probability of not rolling a three is equal to one minus the probability of rolling a three. So that means probability of not rolling a three is equal to one minus one out of six or five out of six, exactly what we found on the previous slide. And then similarly by rearranging, we can calculate the probability of rolling a three by doing one minus the probability of not rolling a three. So one minus five over six, which equals one over six, and that's all because rolling a three and not rolling a three, the probabilities add up to one. So what the complement rule is telling you is because an event and its complement are equal to one, if you want to find the probability of event A not occurring, it's one minus the exact opposite happening. It's one minus the probability of A occurring. If you want to find the probability of A occurring, it's one minus its opposite occurring. So the probability of something is one minus its opposite, one minus its opposite, one minus its opposite probability. That's the golden rule. That's what the complement rule is telling you. So if you look at event diagram for the complement of event A, if my rectangle, which is enclosed and filled with red, if that represents my entire sample space, and if my inner circle represents A, it's this red space on the outside that represents the opposite of A, everything that is not in circle A but is still in the sample space. That's my A bar. So in our first complementary event example, I have a recent survey of 1,065 students where 192 admitted to skipping a class within the past year for no reason. So this is an example we did when we learned about basic probabilities. We're now going to put a little bit of a twist on it. So we're going to calculate the probability a student skips class. That's what we're looking at the outcome, the student skips class. So probability student skips class is equal to out of 1,065 students, how many skip class were admitted to it anyway? That's 192. Which is equal to 0.4 decimal places 0.1803. Now let's calculate the probability a student does not skip class. This we will use to complement our rule for. So the probability a student does not skip class, the probability student does not skip class, is equal to 1 minus the probability a student does skip class, or 1 minus 0.1803. Which is going to equal 0.8197. So those are my two probabilities here. My answer in a, I used it to find my answer for part b. You're like, well, the complement rule doesn't really save us much time. We easily could have taken 1065 and subtracted 192 to be able to get my numerator for students that do not skip class. Well, you're going to learn in a minute that the complement rule is highly beneficial to you. So to show this to you, we have the probability of at least one. The probability of at least one is finding the probability that among several trials, we get at least one of some specified events. So maybe you flip a coin, and you want to find the probability you get at least one heads, or you get at least one tails, you flip a coin 100 times, you want at least one occurrence of heads, or at least one occurrence of tails. Well, at least one translates to one or more. So the complement of getting at least one item is to say you get no items. Those are the complements of each other. One or more, or at least one, the complement is none, no, zero. Because they're complements, the probability of at least one plus the probability of none is equal to one. After arranging this, I have this great formula I can use now. The probability of at least one is equal to one minus the probability of none. So the probability of getting at least one heads when you flip a coin 100 times is equal to one minus the probability of getting no heads when you flip a coin 100 times. Let's use this rule. A company supplies DVDs in lots of 48, and they have a reported defect rate of 0.5%. So that means the probability of a disc is defective, defective now is 0.005. It follows that the probability of a disc being good is 0.995. What is the probability of getting at least one, there's that keyword, at least one defective disc in a lot of 48. So the probability we have at least one defective out of the 48 is equal to one minus the probability that none are defective of the 48. So I need to work a little bit here, finding the probability that none are defective of the 48. So what's going to happen here is I need to calculate the probability that none of the 48 are defective off to the side here. So that means finding the probability that all of the 48 are good, that they have no issue. That means you take your first DVD. What's the probability that first DVD will be good? Well the probability of a single disc being good is 0.995. What about your second disc? Also 0.995. You continue to do that until you get through all 48 discs. So what is 0.995 to the 48's power? That's what we want to calculate here, 0.995 to the 48 power. That's 0.7862. So the probability that at least one disc is defective is one minus the probability that none are defective. So that's one minus 0.7862, which is equal to 0.2138. That's the probability that at least one disc will be defective. There's about a 21% chance that at least one of the disc will be defective in a lot of 48. So once again we use the complement rule. We said if we have at least one, it means that the probability, the calculation one will be one minus the probability of none. To find the probability that none of the 48 are defective, it's like saying what's the probability all of the 48 are good. Which means we multiply the probability of a single disc being good with itself 48 times. And so we calculated it was 0.995 to the 48's power. It's 0.7862. We did one minus 0.7862 to get 0.2138. So what is the probability there will be at least one girl if a couple wants to have four children? Assume boys and girls are equally likely. So we have probability of at least one girl. That's what we're trying to calculate. So that will be, think about what that is, one minus the exact opposite happening. One minus the probability of no girls out of the four children, no girls. No girls is the same thing as saying all boys. So the probability of all boys is equal to, well when you have four children there's two times two times two times two possibilities or 16 possible outcomes. Because child one you can have either a boy or girl that's two outcomes there, child two, boy or girl, child three, boy or girl, child four, boy or girl. Two times two times two times two gives you 16. So the probability of getting all boys out of the 16 outcomes how many are all boys? Well without listing them you should know that the only one is when you get four boys right in a row. The probability of at least one girl is one minus the probability of getting all boys one minus one over 16 and one minus one over 16 actually will give you 15 over 16 and as a decimal that will be 0.9375. So that's using the compliment rule for this question. The only other way to do this question would be to write out all 16 possible outcomes of the four children and to look at how many have at least one girl. But to me it's much easier to use the compliment rule here. That's my opinion. Well anyway, I hope you enjoyed. Thanks for watching.