 Hello there. In this video we will discuss the multiplication rule. Essentially what we'll be doing with this concept is we'll take two basic probability calculations and we're multiplying them together. So the multiplication rule is the probability that sum of it a occurs in the first trial and sum of it b occurs in a second trial. But you have to take into account since these two trials are occurring consecutively, if the outcome of the first event a somehow affects the probability of the second event b, it is important to adjust the probability of b. You need to adjust the probability of that second event happening to reflect the occurrence of a happening. So for instance if you're drawing cards from a deck of cards, there's 52 cards in a deck. Well if I'm drawing a card on the first trial and I don't put it back in the deck, when I go to draw a card from the second or for the second trial there's actually one less card. There's 51 cards in that deck now. So that's what I'm talking about when it says take into account what's happening from the first event when you calculate the probability of the second. Think about that deck of cards. One less card on the second trial assuming you didn't replace the first card you pulled. So the notation you'll see is p for probability, open parentheses a and b, that's a event a occurring on the first trial, event b occurring on the second trial, close parentheses, that's how we represent the multiplication rule. So just a little bit of notation. If you see the probability of b vertical line a, p probability of b vertical line a, it means the probability of event a occurring after event a has already occurred. So this vertical line means probability of b taking into account a has occurred. Basically what I have on the screen, I just rewrote it in a little bit different way. So that means finding the probability of b but taking into account what happened in the first trial with event a. So the formal multiplication rule is when you're calculating an a and probability, event a occurring on the first trial and event b occurring on the second trial, you calculate the probability of a occurring just like you normally would basic probability calculation. But then you multiply it by, you multiply that first probability by the probability that b occurs. Remember that vertical line followed by a means probability of b. Given that a has occurred, that means calculate the probability of b and then take into account a has occurred event a has occurred. That's the formal multiplication rule. There's also an intuitive multiplication rule that's a little bit less heavy on the notation. If events a and b cannot occur at the same time in any simple event, you will literally multiply there's that word, the probability of event a by the probability of event b. But be sure that the probability of event b, be sure that the probability of event b takes into consideration the fact that event a has already occurred. That's the most important thing what I just circled. Make sure that the probability of your second event b takes into consideration the fact that event a has already occurred. Remember, think about that deck of cards. You had 52 on the first trial and it went down to 51 on the second trial because we're assuming you didn't replace the card back into the deck. Suppose 50 drug test results are given from people who use drugs. So I have 44 positive results, 6 negative results. If 2, 2 of the 50 subjects are randomly selected without replacement, there's a reason why the word is bold without replacement, find the probability that the first person tested positive and the second person tested negative. So the probability that I'm going to calculate is the probability that my first test result is positive and my second test result is negative. So the probability that the first is positive and the second test result that I pull is negative. So literally what I have to do is I have to calculate the probability of a test result being positive and I'm going to multiply that by the probability that a test result will be negative. Given that the first result was positive. Remember without replacement so that means we're going to have fewer test results to pick from. So probability that the first that you get a positive on the first pool times the probability you get a negative on the second pool. Given that you had that positive pool on the first draw. Alright so out of my 50 items, how many were positive? 44 times. And then when I go through and I calculate the probability of getting a negative result, how many results are negative? 6. And how many results do I have to pick from? Because this is without replacement. Remember without replacement. Well there's only 49 to pick from and that's what it means by finding the probability of getting a negative result given that the first one was positive. It means that I had 49 test results to pick from. Alright so now what's going to happen is we are going to literally multiply these two fractions together. That means 44 times 6 which will give you 264. And then you have 50 times 49 which is 2450. So as a decimal to three places that's .108. So that's an example of using the multiplication rule. The probability of the second event or second trial was altered slightly because of what happened in the first. That's where the 49 came from. Well let's see. If two of the 50 subjects are randomly selected with replacement, find the probability of the first person tested positive and second person tested negative. So the probability of the first test results positive and that the second test results negative. So I had to find the probability of the first results positive and I'm going to multiply it by the probability that the second result is negative. There's no need to consider, no need to consider what happened in the first trial since we have replacement. So basically you're going in and you're picking out one test result and you're throwing it back into the pool. That's what we mean by with replacement. So as a result when I do my calculation here the probability of a positive test result is indeed 44 out of 50 and the probability of a negative test result is there's six negative results out of 50. We do not change the number of test results available because we are replacing what happens in the first trial. We'll talk about this more in a little bit. So 44 times six is 264 and 50 times 50 is actually 2500. So I took two basic probabilities multiply them together. So the fact of if you have replacement or you don't have replacement does impact your answer because my answer here is point one of six. So let's talk more about whether or not you should consider what happens on the first trial or not. So this brings us to the discussion on events being said to be independent. Well events are independent if the occurrence of one has no effect on the probability of any other event. So that means whatever happens with this event does not impact the probability of another event. So two events A and B are independent if any of the following is true. So if you are told to calculate the probability of event A occurring given that B has occurred this is called a conditional probability but if you're told to calculate the probability that event A occurs given that B occurs since the two events are independent so I'll say independent so whatever happens with event B does not affect A. So it's just like calculating what is the probability of A and similarly if you want to calculate the probability of B given that event A occurred on another trial well because these two events are independent A does not affect B at all. So it's just like calculating what is the probability of B this little condition this little what happened in the first trial drops off because of independence. So therefore when you calculate the probability of A occurring and then event B occurring it means you just calculate probability of A probability of B and then you multiply the two probabilities together. So it's literally two basic probability calculations with no strings attached. That's the cool part about independence. So in general the probability of any sequence of independent events is literally the product of their corresponding probabilities. So if you had 10 independent events and you wanted to calculate their probabilities you just multiply 10 probabilities together that's what we mean by that and events are said to be independent when an experiment is performed with replacement. So when you see those words with replacement that means you do have independence whatever happens on the first trial does not affect the next trial and events are said to not be independent when you have without replacement. So like when you have a deck of cards there's one less card in the deck if you don't replace it so it impacts the probability of future events. So if A and B are not independent we say they're dependent. So that's the opposite of independent is dependent. So one little other strings attached thing is the five percent guideline for cumbersome calculations. If a sample size remember that's how many things you're pulling from a total group is no more than five percent of the size of the population that's everything under consideration. You can assume even if it's logically not the case but because of this less no more than or less than five percent of the size of the population you can assume or treat the selections as being independent and we'll see an example of that momentarily. So determine whether the two events are independent or dependent then calculate a probability of event A occurring and B occurring is such an event unlikely. So a student randomly picks an answer on the first question of a multiple choice to choice test and chooses D. So that's my that's my first event A and then my second event B is a student randomly picks an answer on the second question and chooses C. So chooses D occurs first chooses C will occur second. So the big question is does this student picking D on the first question impact the probability that the student will choose C on the next question. In other words since there's four answer choices will there still be a one in four chance in either case and the answer is yes. So we say we are independent and that's because choosing D on the first question does not impact the probability of choosing C on the second question. So if I wanted to calculate the probability of choosing C and probability of choosing D on the second question it's literally going to be saying okay let's take the probability that the student chooses C and let's multiply it by the probability the student will choose D on the second question. So whenever I get to the first question if I'm the student what's the probability I'll choose C out of four answer choices how many are C? Just one. When I get to the second question out of four answer choices how many are D? Just one. So you end up getting 1 over 16 which is actually going to give you 0.0625 and then they want to know if this event is unlikely. The definition of unlikely is if the probability is less than 0.05 so I'll say since 0.0625 is greater than 0.05 the event is not unlikely. It's got to be less than 0.05 to be unlikely. So then answers all the questions here. All right so now I have exam grades from two classes. Two of the 60 students are randomly selected. Assume the selections are made without replacement really powerful word here without replacement. So that means we are not independent. That means whatever happens with my second grade pool needs to take into account what happens from the first. What is the probability the two students both made B's? All right so I want to calculate the probability of getting a B on my first grade pool or grade selection and getting a B again on my second grade selection. So these are not independent. So here's what happens. You go through and you calculate the probability of first getting a B. So I have what 60 students total. How many students have B's? 19. So when you go to pick your first grade the probability or likelihood of getting a B is 19 out of 60. When you go to pick your second grade because you do not have replacement you do not put that grade of B back into the pool of possible values or possible things to pick. For that reason there's now 59. Not only that because the first trial was also a B and you wanted the probability of also getting a B on the second trial because you do not have replacement because the events are not independent you also have one less B in the mix. So the 59 is because there is one less grade because you didn't replace the one you picked the first time and the 18 is because there's one less B because that first pool we are looking at it being a B. Multiplying the probabilities together you're going to get 342 over 3540 which is actually going to give you 0.097 that is the probability. What about a person that has to be at work early the next morning and his alarm clock has a 12% failure rate? What is the probability his alarm clock will not work the next morning? Well if his alarm clock has a 12% failure rate the probability of it not working will be 0.12. Like if there's a 12% chance of me not waking up I think I probably better fix the issue. So what if I set two alarm clocks? What is the probability they both fail? So the probability first one fails and then probability the next one fails. Well are we independent? If one alarm clock doesn't go off will it impact the probability that the next one won't go off? No it doesn't these two events are independent the two alarm clocks operate separately. So literally all I have to do is take the probability the first alarm clock fails and multiply it by the probability the second alarm clock fails. So 0.12 times 0.12 and it turns out that we go down to actually being 0.0144. So a 1.5% chance of our alarm clock failing. So that's a little bit better but what if I'm paranoid I'm like I want to be absolutely certain I have at least one alarm clock go off. What if I set three independent alarm clocks? So literally multiply the probability of failing three times. So that's literally 0.12 times 0.12 times 0.12 which will actually give you 0.0017. So we went from a 12% chance of no alarm clock going off to a 1.5% chance to practically a 0% chance of an alarm clock not working. So I would say yes. Adding that second and third alarm clock does help improve reliability. So that's kind of a cool take on things there. Next in a recent survey of 1065 students 192 admitted to skipping a class within the past year for no reason. If one of the surveyed students is randomly selected what is the probability that the student did not skip class within the past year? So 192 skipped. How many did not skip? Well that would be 873. 1065 minus the 192. So 873 did not skip. What is the probability that the student did not skip class within the past year? The probability the student did not skip would be out of 1065. There's 873 that did not skip. So that's the numerator. You divide the two to get 0.8197. If 24 of the surveyed students are randomly selected within without replacement what is the probability none of them skipped class within the past year for no reason? Should the 5% guideline be used? Remember what the 5% guideline says. It says that if the sample size which is 24 in this case is less than 5% of the population size we can assume the events to be independent. So let's look at this. What is 5% of my population? What is 5% of 1065? So it's about I would say 53. Since my sample size is less than 53 we can assume independence. So this is 24 students. I'm finding the probability that 24 did not skip class. So if you think about this this means okay student one what's the probability that student did not skip? Student two probability that one did not skip. Student three probability that one did not skip. All the way to the 24 student probability that one did not skip. You're literally taking the probability of not skipping which is 0.8197 and you're multiplying it with itself 24 times. Remember that's 24 times so 0.8197 to the 24th power exponents represent repeated multiplication by the same number and you actually get 0.0085. So that's the probability that 24 students that out of 24 students none of them would skip. Once again we use the cumbersome guy 5% guideline for independence. So that's all I have for now. Thanks for watching.