 Hi there, in this video we are going to discuss the addition rule. What the addition rule basically is, is you're looking at multiple outcomes as being the favorable outcome in an experiment. So that being said, first off, by definition a compound event is any event combining two or more simple events, such as flipping a coin and rolling a die. The two can be brought together into the same experiment, but once again they're both two different simple events there. Another definition, we call events A and B mutually exclusive or disjoint if they cannot occur at the same time. So if I have two events, event A, event B, if they have overlap of their outcomes, some of their outcomes are the same, then they are not disjoint. They have overlap. Now if events A and B have no overlap, there's no outcomes that are the same, then they are disjoint. They have nothing in common. So disjoint events cannot happen at the same time. Disjoint, mutually exclusive mean the same thing. So here I have two events, are they mutually exclusive? Randomly selecting someone who owns a boat, randomly selecting someone whose favorite color is blue. Can that happen at the same time? Can I go around and pick one person and they both own a boat and their favorite color is blue? The answer to that is yes. So they can occur at the same time, therefore they are not mutually exclusive. Not mutually exclusive because they can happen at the same exact time. And B, I have randomly selecting someone who likes math and randomly selecting someone who dislikes math. Can that happen at the same time? Can someone both like and dislike something at the same time? Well at a particular moment of time, an instantaneous moment of time, the answer is no. So they cannot occur at the same time. That being said, these events are mutually exclusive or disjoint. So that's mutually exclusive events or disjoint events. Now the addition rule is used to find the probabilities of the form, the probability of event A occurring or event B occurring. So that's the probability that either event A occurs, event B occurs, or they both occur. So the keyword here and when you're looking at your questions is to find the word or. So it's this one, that one, or both of them. That's what the or probability is, and it's the addition rule. The formal addition rule states to calculate the probability of A or B, you take the probability of A, you add the probability of B, and you subtract the probability of A and B occurring at the same time, that overlap. The reason why this is important is because think about event diagram that has some overlap in the middle. You have outcomes A on the left circle and outcomes B on the right circle. Think about what happens when you add up the probability of A occurring, when you take circle A, and then you add to it the probability of event B occurring, circle B. Notice where I shaded twice. My overlap in the circles has been added twice to my probability it was double counted. So that's the reason why in the addition rule you have to subtract the overlap here, the probability of A and B occurring at the same time because that area was counted twice. So here's the intuitive rule, is to find the sum of the number of ways event A can occur and the number of ways event B can occur, but be sure to add in such a way that outcomes are only counted once. So you add in such a way that outcomes are only counted once, there's no double counting. So this sum divided by the total number of outcomes is how you find the or probability. So I have an employer that drug test its employees. If an employee is randomly selected, what is the probability the employee test positive or negative for drugs? Instead of probability an employee test positive or test negative for drugs? So how many people total do I have? I believe it's the sum is 99, 41 plus 9 plus 10 plus 39. So out of 99 people, how many tested positive? How many did negative? Well 41 and 9, those are my positive results. What about my negative results? 10 and 39, those are my negative. So we actually have everything. We have 41 plus 9 plus 10 plus 39. The results are all either positive or negative. So you get 99 over 99 which is 1. Let's do one that's perhaps a little bit more exciting here. The data summarizes results from 870. So that's my sample size here. My total group is 870 pedestrian deaths that were caused by accidents. If one of the deaths is selected at random, find the probability the pedestrian was intoxicated or the driver was intoxicated. So I'm finding the probability that the pedestrian was intoxicated or or the driver was intoxicated. So any of these data values where the pedestrian was intoxicated or the driver was intoxicated will be included in our favorable outcome. So out of 870 pedestrian deaths, how many did we have for the pedestrian intoxicated? Well that's the 63 and the 261. Then what about the driver intoxicated? Driver intoxicated is 63 again but we already have it circled. So we're already counting it. Don't count it again. And then the 45. So that's when the driver was intoxicated and then we also have when the pedestrian was intoxicated. Any of those stipulations is a favorable outcome. So we have 63 plus 261 plus 45. At the end of the day you get 369 out of 870. And we'll round to let's say four decimal places. So we'll divide. We get 0.4241. 0.4241. 0.4241. So or make sure you don't double-count anything. What about another example? A survey asked 75 people in the 18 to 21 age bracket a question which 49 responded and 26 refused to respond. 200 people in the 22 to 30 age bracket were contacted, 172 responded and 28 refused to respond. Suppose one of the 275 people is randomly selected, find the probability of getting someone in the 18 to 21 age bracket or someone who has responded. Yeah. So many words. Well first off the probability I'm calculating is someone being 18 to 21 or someone that responded. I need to take my data and I want to organize it in a table because that's the best way to work out these questions. So it looks like responses are broken up by age. I have 18 to 21 and I have 22 to 30 and they're also broken up into those that responded and those that did not. So I have responded and then I have those that refused to respond or did not respond. So let's take the numbers from all these words and break them up into the table. So there are 75 people in the 18 to 21 age bracket 49 responded and 26 refused. So 49 responded this is for the 18 to 21 age bracket 49 responded 26 refused. In the 22 to 30 age bracket all for those 200 people 172 responded and 28 refused. So those are my numbers that's what I put in my table. Any value that is in the 18 to 21 category that would be 49 that would be 26 and any value that's in the responded category that would be 49 again but don't double count and 172. So out of the all the people 275 favorable outcomes for 49 plus 172 plus 26 add together those numbers and you actually end up getting 247 out of 275 and that's going to give you 0.8982 0.8982 that's your answer here. So it does help to organize your data into a contingency table otherwise it can turn into a notational nightmare. We want to keep this as real and practical and as visual as possible. So just as a quick reminder the addition rule versus the multiplication rule in the multiplication rule they use the word and and that means you're finding the probability of event a occurring you're finding the probability of event b occurring take into account what happened first and you're multiplying the two probabilities together so multiplication really do just that you multiply it uses the word and and the addition rule use the word or you find the probability of event a occurring or event b occurring and it suggests addition that's why it's called the addition rule. So you add together the probabilities of each individual event in such a way that every outcome is counted only once you do not want to double count. So that's distinguishing between the two rules. So other than that that's all I have for now. I hope you enjoyed thanks for watching.