 binomial cumulative distribution and you absolutely want your graphing calculators handy and once again if you don't have a graphing calculator you want to fess up and swap and give me one. I thought I yelled at you last day. I yelled at you again today. It's an actual phone. Who carries a phone anymore? It's a dumb phone. It's not a smart phone. It's a dumb phone. Recall. So here's what we've done so far and Brett can I pick on you as an athlete because I want to show you some of where this is used in athletics. So example one. Brett shoots free throws at we'll make you a half decent free throw shooter. We'll go 68% which for high school is tolerable. Okay. I think Mr. Goulet's goal is over 70. Is that correct if I recall? Team goal, right? It makes Sam DeHenepi's career 84 look pretty all that much more impressive if you look at the career stats. He was a machine. A. Find the probability that he hits exactly seven out of his next 10. Ready Jasmine? Pick on you again in love. How many shots is he taking grand total? Ten. Tree? No. Two. Maybe. No. So we would say this. N equals 10. N was the number of trials. Now the variable that we used for the number of successes was X. How many does Brett want to go in? Seven. You're not here for the analysis though. We're saying we want to find out the odds of hitting seven. And we use the letter P for the probability of success. What's the probability of Brett experiencing success? Point six eight. And then we use the letter Q for the probability of failure. What's the probability of failure? Do the arithmetic compliment? By the way, I think on your formula sheet it says this Q equals one minus P. But I'm hoping you kind of know that one. It's point three two. Do you guys want to use the longer choose version or do you want to use the shorter built-in graphing calculator? Okay. So what will we do? This is what you would show for work. And I'll hand write it so you can see. This is what I would expect. I would expect you to do a little list like this, although you don't have to. I guess you can go straight to the function. But those of you who have physics with me know I'm deficking, right? Listing my data. I would expect you to write binome PDF. And you do need to have the syntax memorized. Do you remember what the syntax went where? And I'll give you a hint. It's alphabetical. N, P, X. Sorry, Q never shows up. Okay. If you're looking for a dumb way to remember, it's going to be N comma P comma. What are the odds of him hitting seven? Exactly seven, not eight, not six, but exactly seven. Now we need to find where that function was. Okay. You need to memorize where this is. If you're going to go with the graphing calculator route, I will not show you where it is. I will just laugh at you. Now where was it? It's under the distribution function. And by the way, you can see there's a whole bunch of different types of probability distributions. There are some where the what if there's more than two outcomes? Yeah, that's one of those that's built in there. What if there's the odds are changing? That's one of those that's built in there. Here in math 12, we only look at the binomial 10 comma 0.68 comma seven. He's got a 0.2644. 0.2644. By the way, I won't do this on the notes, but Brett, here's the good news. What are the odds of you going over? Pretty slim. Here's the bad news. What are the odds of you going 10 for? Well, 2%. It could happen. You never know. Next question. Find the probability that he hits exactly 14 out of 20. Will it be the same as seven out of 10? Why not? Now you have 20 possible outcomes that are spreading out that 100% grand total. There's way more possible outcomes. The odds of any one outcome is going to go down. So you can't simply say, whoa, the fraction reduces 14 out of 20. You seven out of 10. Same odds. No. And I'll give you an example that makes it clear. What if he was taking a million free throws? What would the odds be of him hitting exactly 140,000? No more, no less. Pretty slim. The odds would definitely change. So this would be binome PDF of what's and this time? How many trials? 20 comma 0.68 comma 14. Not as likely because now you're aiming for a smaller target. 0.1881. Is there anybody that's not able to get this on your calculator? There's your chance to ask. All good? All right. Let's go back to 10 free throws. Here is a much more relevant question because somebody earlier said, well, he wants to hit all of them. I agree. Technically, he's not aiming for a specific number. However, your team goal was always 70%. So seven out of 10. So here's what I'm going to ask. What is the probability that he hits seven or less out of 10? Now, Matt here said at least seven. Actually, your calculators can't find seven or higher. What they can find is seven or less. Your calculators have or less built into them. Although, wait a minute, if 70% is acceptable, then Brett, what I want to figure out, sorry, cross out the seven, I want to figure out six or less because the compliment will be seven or more. Will it not? The opposite of six or less will be seven, eight, nine or 10. And that was Mr. Goulet's cut off for your team. So cross out the six, seven, make it a six. And I did that on purpose because that's the most common mistake people make by the way. They say, oh, I want to figure out seven or more. I'll go seven or less. Well, one minus seven or less. No, no, one minus six or less. Your calculator has this built in. You could, what if instead though, we're talking about a provincial exam that has 60 questions and you want to find out the odds of you passing. Do you really want to go 30 or 31 or 32 or 33 or 34? I agree. For small ones, I can type them in individually, but I'll be, the more interesting one are the bigger questions. Like certainly exams. Yeah. Oh yeah. I'd kind of like to know, Mr. Goulet. Sure. Right. Or for a whole season, how many free throws do I shoot on average in a season? I'd like to, I'd like to, for the whole season. So yes, here, this is why I said I actually don't use those curly brackets all that much. I use a built in but somewhat obscure function on my calculator. Okay. Okay. The binome CDF and I've capitalized the C to make it stand out. It's the one right below the binome PDF on your calculator. So you can find it already because just go one down further of N comma P comma X. Oh, and I got an L in there for some reason. I don't know where that L came from. The binome CDF and I capitalize the C to make it stand out of that will calculate the odds of X or less and add them. And it stands for binomial. That's what the B stands for. The capital C stands for cumulative distribution function. And you guys have been in school long enough. What does cumulative mean? Includes everything. Everything below and including zero. So if I want to find the probability that he hit six or less. So six or less includes includes six or five or four or three or two or one or zero. And Jen, I could do that with the brackets. Go ahead and then use that sum feature. Go ahead. Don't forget the zero though. That's the one kids often forget. He could go over. Thankfully, there wasn't much of a chance of that. But he could or use binome. And once again, I'll capitalize the C. So it stands out CDF of 10 trials. What was the odds of success? Point six eight comma using the C that will calculate six or five or four or three or two or one or zero and add them up for me. So what are the odds of him going six or less? Point 404. Give it to me to four decimal places, please. Point 404. Anybody else double check? Yes. Oh, I'm typing. So why not? 0.4044. Now, the question Brett wants to know then is D, what is the probability that he hits seven or more? Another way to phrase that Megan is at least seven. That means seven or more. You cannot calculate seven or more directly because the CDF always takes the number that you put in and goes less than. In other words, if you want to do it or more question, you have to use the compliment. This is one minus six or less. And the nice thing is you can actually type this in one minute. Let's try that again. Mr. Do it. One minus six or less means I want to use the cumulative second function distribution. Ten shots comma 0.68 comma six or less. One minus that should give me seven or more. And hopefully you can see certainly in sports where managers and things strategic people who are trying to plan their strategy would be interested in this big ones, especially football and baseball. They don't do stats as much in basketball, but football and baseball, they have really bought into the whole stats thing. And I think I've mentioned to you, you guys know the other guy who's a big stats nut, right? Dave Nones, the Canucks general manager, big time into money ball. This is why you're seeing he's trading for people often that you've never heard of, but darn if they aren't fitting in quite well because he's analyzing them statistically and he's saying, I've got a statistical model of the Canucks, here's where we're lagging. I don't care what else this person has. I want someone that will cover up this particular statistic. And the rest will take care of itself. It's true. 0.5956. Yeah, you know what though, I'm going to tell you, I'll probably give you this question right away without walking you through the ABC and D. Okay. The one you guys are wondering is provincial exams. So all of you are writing English 12. Does anybody know how many multiple choice questions are there on the English 12 exam? No one's looked at it yet? 14 or 16, that's it? It's a lot of English. Okay. Well, math. Okay. Example two. On the math provincial, is it example two? I think yes. I hope. Example two, on the math provincial, there are 44 multiple choice questions with four possible answers for each. In other words, what are the odds of guessing right on any one question? One out of four. What is the probability of passing by guessing? There's 44 questions. What's the pass? 22 or 23 or 24. Jen, this is where the bracket adding them all up method could be totally could be done. You look tedious. Or 25, or 20, or 20, or 20, okay. But I would argue though, Jen, this is to me is a relevant interesting question. Like this is some of the stuff you probably wondered over the years when it comes to big final exams and tests. So I think it's going to be, I want the probability of 22 or more. I can't do using the binomial or more. The binomial CDF only does or less. So it's going to be one minus the probability of what? 21 or less. Which is going to be one minus. And the big problem, Jordan, is kids, students get PDF and CDF mixed up. So I'm going to say it again. What does the letter C stand for? Cumulative, which implies all of them. P stands for probability of particular example. So I want here or less CDF, binome CDF of how many trials grand total? 44 comma. What are the odds of success? 1 out of 4 or 0.25 comma. And I want 21 or less. I kind of like to know. I doubt it's going to be very high. Matt's saying even when I made it a percent by time thing by 100, it's still really small. Okay. Sure. As a percent, so three one hundredths of a percent. You know what the moral of study is, Jen? And you guys have all figured that out because your academics do study. Study. Study. Study. You can do some very nice questions though with trying to remember was it this class? No, different class. Another class asked me, so what are the odds of it wasn't the connex? Pardon me? Oh, can you slide the calculator to the right? In fact, why don't I do this and I'll write down those. 3.08 times 10 to negative 4. 3.08 times 10 to the negative 4 or 0.03 percent. So connex. How many games is the connex series? Okay. So a good question would be in seven games, what's the probability of the connex win for? If you know their winning percentage, you would go by known PDF, seven comma, whatever their winning percentages, comma four. That's if they win four out of seven, they could also win or four out of six or four out of five or four out of four. But there, if you want to figure out, if you're gambling, if you want to figure out what kind of odds Vegas is going to put out, I'm willing to bet they calculate it somewhat that way. And then they'll add a fudge factor to make sure that the odds are they'll make a profit no matter what. Is that all right? Binome CDF and Binome PDF? We're done probability. I got a couple of more neat and nerdy things that I will talk about, but what's your homework? You got it down? Okay. If you go to page 491 homework, number three, number four. Oh, by the way, what if I wanted to figure out the odds of Brett hitting between three and seven shots? What I do that, seven or less, take away two or less, whatever's left is three, four, five, six, seven. You subtract two CDFs. Let me figure that out. But in other words, Jen, I can use that CDF feature really to get any range of numbers that I want to if I'm a little bit clever. So three and four, oh, penalty shots, six, eight. And then I have for you also a take home quiz. And I'm going to give you the great big probability unit review, which I deliberately didn't hand out until right now. You can start working on that as well. Your test is not next week, it's the week after. I see you twice next week. One is going to be going over the great big probability quiz. And one is going to be Q&A and a few other things. You guys are actually a class ahead of my other group. So