 Welcome back. In this video, we will get a little bit more practice using the binomial probability distribution calculation formula, and then we'll talk a little bit more about the mean and standard deviation of binomial probability distributions. In the example I have that when someone buys an airline ticket, there is a .0888 probability that the person will not show up for the flight. A certain jet can seat 11 passengers. Is it wise for the company to book 13 passengers for the flight? Because after all, there is a chance that people won't show, but is it a high enough chance? So what we're going to do here is what I'm calculating or what I'm interested in would be the probability of there not being enough room or that the flight is overbooked, meaning more people show up than what we have seats. So that would be the probability that if you only have 11 room for 11 passengers, that means the probability that 12 or 13 show up. This is the context which I'm going to approach this question. Let's find the probability that the flight gets overbooked, meaning 12 or 13 people show up. So because this is the binomial probability formula question, obviously we're going to be using the probability, binomial probability calculation formula. So what I'm going to do is to find the binomial experiment. And in my experiment, a success is somebody shows up. A success is somebody shows up. So what's the probability someone shows up? Well the probability someone doesn't show up is .0888. So the probability someone does show up is one minus that because someone shows up, someone doesn't show up their compliments. So one minus .0888. And you're actually going to get .9112. So when I go to Google Sheets to do this calculation, we use our spreadsheet as a form of technology to do this calculation for us. The number of trials in is actually going to be 13 because you're looking at out of the 13 passengers how many show up. The probability someone shows up is .9112. I found that by taking one minus the probability someone doesn't show up. And then I need to identify my lower bound and my upper bound. Well if you're interested in either 12 or 13 people showing up, that means the lower bound of people, the number of people you're interested in is 12 and the upper bound is 13. So literally I'm going to go to my Google Sheets document and type in this information. So Google Sheets will use the compute tab and will be in the binomial region. So the number of trials that we have is going to be 13. The probability someone shows up or the probability of a success is .9112, lower bound 12, upper bound 13. And we get about .6767. Your homework will tell you what to round to. So we'll do four decimal places for this one. .6767. So the probability that flight's overbooked is .6767. That means about a 67 and a half percent chance that the flight will be overbooked. So if you're an advisor to this airline company, would you advise them to book 13 passengers on an 11 passenger flight? I would say no. There is a good chance the flight will be overbooked. Meaning someone's going to be standing or someone's going to have to stay behind. And that never ends well. So let's be smart about our business decisions, right? So they asked me, is it wise to book 13 passengers so I calculated the probability that the flight would be overbooked? Meaning if 12 or 13 passengers are booked and show up. Then I got a relatively high probability. That's about 67 and a half percent. Alright so we previously in the other video we talked about the mean and standard deviation of probability distributions. Well there's good news. When the probability distributions binomial there's shortcuts we can use to find the mean and standard deviation. Literally if a random variable x has a binomial distribution we say that x is binomial with a mean mu standard deviation sigma. If you take any statistics courses after this one you might see that formal theoretical notation. But all you need to know for this class is to find the mean of a binomial distribution mu is equal to n times p, the number of trials times the probability of a success. The standard deviation is n times p times q and you take the square root of that product. And then sigma squared is just n times p times q. So remember what these variables or these values stand for. n is the number of trials, p is the probability of a success. Remember you define what the success is for each experiment and q is the probability of a failure. Really important q is always 1 minus its opposite 1 minus p. Failure is the opposite of success. So what the range rule of thumb states is that usual values lie within two standard deviations of the mean. So the maximum usual value is you take the mean add the standard deviation to it twice. It's really important that you always round down. Because when you're talking about discrete distributions, which is what a binomial distribution is, you're talking about whole number values. So if the maximum usual value is 13.7, 14 exceeds that. So that's why you round 13.7 down to 13. 13 is the most practical maximum number you could have. Minimum usual value, you'll literally take the mean and subtract the standard deviation twice or subtract two times the standard deviation. You always round this value up. So if you get a minimum usual value of like 7.2, that means seven is far too low. So eight is the first actual possible value that could be considered usual. 7.2, you got to round it up. So keep those rounding rules in mind. Please keep them handy. So McDonald's has a 95% recognition rate. Surprise. A special focus group consists of 12 randomly selected adults. For such a group, find the mean standard deviation, minimum usual value and maximum usual value. So all I need to do is identify my N, my P, my Q and I can find the mean and standard deviation. Well, my number of trials, which is number of people here would be 12. I'll go ahead and be official. A success is recognizing McDonald's. So that means what is the probability of a success? What is the probability of recognizing McDonald's? It's actually 0.95. So what's the probability of a failure of not recognizing McDonald's? That's one minus 0.95. So we've identified the key ingredients to now find the mean and the standard deviation. The mean is N times P, 12 times 0.95. And you actually end up getting 11.4. My standard deviation is the square root of N times P times Q. N times P times Q. So I have literally 12 times 0.95 times 0.05. You'll use a calculator to calculate this and then you'll take the square root of that product that the three numbers multiply together. You'll actually get about 0.755. So we've answered the first two parts of the question. Let's go ahead and find our minimum and maximum usual value now. So this is my minimum usual value. It's found by taking the mean and subtracting 2 times the standard deviation. That's 11.4 minus 2 times 0.755. So 2 times 0.755, take that result and subtract it from 11.4. You'll actually end up getting 9.89. And you will always take the minimum value. You will always round up. Always round up the minimum usual value. So 10. Alright, what about the maximum usual value? Feel free to pause and try it out if you want. The maximum usual value. Well, I literally take mute and I add the standard deviation to it twice. 11.4 plus 2 times 0.755. Perform that calculation and you'll get 12.91. So that means if something's 13, that's too high. For maximum usual value, you always round down. Besides, we can at most have only 12 people total. So another reason why you should bring the maximum usual value down to 12. Not only should you round down, but you should also never exceed your number of trials. Just a little bit of warning there. A little bit out of the box thinking. So the minimum and maximum usual value. Sometimes you'll have to put it in interval notation brackets with comma separating the numbers 10 comma 12. That may be how they ask you to do the response or they may just ask you to type in 10 type in 12 and you're done. Let's practice a little bit more. So we have unprepared students that are given a 10 question pop quiz. The quiz is multiple choice and find the mean and standard deviation for the number of answers or correct answers a student obtains. Would it be unusual for a student to pass by guessing? Passing means they get at least six of the 10 questions correct. So a student goes in to take the test. We'll say a success is getting a question correct. That being said, okay, I have 10 questions or 10 trials. The probability a student gets a question correct when there's four answer choices is one out of four, which we already said for plenty of times that's 0.25 and Q or probability of a failure is one minus P one minus 0.25. Calculate the mean in times P. So that's 10 times 0.25. And that's gonna give you 2.5. What about sigma? What about the square root of N times P times Q? What about the square root of 10 times 0.25 times 0.75? Multiply the three values together then take the square root. 1.369. We are almost there. So now we need to figure out if it's unusual for someone to get at least six correct answers, which means we need to figure out what is the range of the minimum and maximum usual value. So let's find the minimum usual value and let's find the maximum usual value. Let's see if six lies within there. So the minimum usual value is literally your mean minus 2 times 1.369. So that means you're actually going to get negative 0.238 and it does not even make sense to have a negative number of questions correct. So by default 0 is the minimum usual value. Maximum usual value we have 2.5 plus 2 times 1.369. That's 2 times the standard deviation and you actually end up getting 5.238. So that means our cutoff is at 5.238. Our cutoff is at 5 because we always round that maximum usual value down. Always always round down whether it actually does or not always round down. So we get 5. So 6 is not between or equal to 0 and 5. So that's our spread of usual values. That means 6 is not in that interval. That means 6 is unusual. So it is unusual for a student to pass by guessing. So it's not likely. So if you're one of the ones that takes a 10 question multiple choice quiz and you pass it all from guessing good for you. That means it's your lucky day. So 6 is not in the spread of usual values. Therefore it is unusual for a student to pass by guessing because passing means getting at least 6 questions correct. So that's all I have for you for now. Thanks for watching.