Probability using combinations

so you might be wondering why I went off into permutations and combinations in the probability playlist and I think you'll learn in this video so let's say I want to figure out the probability to the problem I'm going to flip a coin eight times and it's a fair coin and I want to figure out the probability of getting exactly three out of eight heads so I say three eight heads but three of my flips are going to be heads and the rest are going to be tails so how do I think about that well let's go back to one of the early definitions we use for probability and that says it's the probability of anything happening is the probability of the number of equally probable events in which what we're saying is true so in which we the number of events the number of events in which we get three heads and exactly three heads we're not saying greater than three heads so four heads won't count and two heads won't count five heads we'll only three heads and then over the total number of equally probable trials so total number of total total number not trials total number of equally possible outcomes I should be using the word outcomes so just with the word outcomes it should be the total number of outcomes in which what we're saying happens so we get three heads over the total of possible outcomes so let's do the bottom part first let's do the bottom part first what are the total possible outcomes if I'm flipping a fair coin eight times well the first time I flip it out either get heads or tails so I get two outcomes and then when I flip it again I get two more outcomes for the second one and then you know how many total outcomes well that's two times two because I'm going to add two in the first two in the second flip and then essentially we multiply two times the number of flips right so that's five six seven eight and that equals two to the eighth so the number of outcomes is just going to be two to the the total number of flips and hopefully that makes sense to you if not you might want to re-watch some of the earlier videos but that's the easy part so there's two to the eighth possible out when you flip a fair coin eight times so how many of those outcomes are going to result in exactly three heads let's think of it this way let's let's give a name to each of our flips let's give a name to them so let me make a little column call these the flips this is my flips column and I could name them anything I could name him Larry Curly moe I could name him the and why would need five more names for them but you know I could name them the seven dwarves or the eight dwarves really could I have eight flips but I could you know I'll just name the I'll number the flipped flips one two three four five six seven eight and I'm the god of probability and essentially I need to just pick three of these flips that are going to result in heads so another way to think about it is these could be eight people and I you know I could pick which of these you know how many ways can I pick three of these people to put into the car or how many ways can I pick three of these people to sit in chairs and it doesn't matter the order that I picked them in right it doesn't matter if I say the people that are going to get in the car are going to be people 1 2 & 3 or if I say 3 2 & 1 or if I say 2 3 & 1 those are all the same combination right so similarly if I'm just picking flips and I have to say okay three of these flips are going to be are going to get into the heads car or we're going to sit on you know heads is like they're sitting there people sitting down I don't want to confuse you too much but essentially I'm just going to choose three things out of the eight so I'm essentially just saying how many combinations can I get where I pick three out of these eight and so that should immediately ring a bell that that we're essentially saying out of eight things eight we're going to choose three and that's our that's you know how many how many combinations of three can we pick of eight and that's we went over in the last video and let's do it with the formula first so let me write the formula up here just so you remember it but I also want to give you the intuition again for the formula so in general we said n choose K that is equal to n factorial over K factorial times n minus K factorial so in this situation it would be that would equal 8 factorial over 3 factorial times what 8 minus K times 5 factorial or another way of writing this this would be 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1 over I'll just write 3 factorial for a second then times 5 times 4 times 3 times 2 times 1 and of course that and that cancel out and all you're left with is 8 times 7 times 6 over 3 factorial and I did this for a reason because I want you to to reget the intuition for at least for this part of the formula that's it just essentially just saying how many permutations can I you know how many ways can I pick three things out of eight and that's essentially saying well before I pick anything I can pick one of eight then I have seven left to pick from for the you know second spot and then I have six left to pick for the third spot right and so that's essentially the number of permutations but since we don't care you know if we picked it what order we pick them in we need to divide by the number of ways we can rearrange three things and that's where the 3 factorial comes from and so we're just hopefully I didn't confuse you but you know if I did you can go back to this formula for the binomial coefficient but it's good to have the intuition and then well once we're at this point we can just calculate this well what's this this is 8 times 7 times 6 over 3 factorial is 3 times 2 times 1 so that's 6 so 6 cancels out so it's 8 times 7 so there's 8 times 7 or what is that 50 56 right for 56 yeah that's equal to 56 so there's 56 different ways to pick 3 things out of 8 or if I have 8 people there's 56 ways of picking 3 people to sit in the car or you know however you want to view it but if I have 8 flips there's 56 ways of picking 3 of those flips to be heads so let's go to our original probability problem what is the probability that I get three out of eight heads well it's the probability it's the number of ways I can pick three out of those eight so it equals 56 over the total number of outcomes right the total number of outcomes is two to the eighth another way I could write that 56 let me unseparated sate times 7 8 times 7 over 2 to the eighth eighth is 2 to the third right let me let me erase some of this not with that color let me erase that erase all of this so I have space I will switch colors for variety use the small pen ok so I'm back alright so 8 is the same thing as 2 to the 3rd times 7 this is all just mathematical simplification but it's useful over to the 8th and so if we just divide both sides put the top then we're in the denominator by 2 to the 3rd this becomes 1 this becomes 2 to the 5th and so it becomes 7 over 32 is that right so it's the if I were to pick 3 out of 8 yep I think that is right and and so what does that turn out to let me get my calculator I have to make careless mistakes let's see my calculator seems to have disappeared let me get it back there it is ok 7 divided by 32 is equal to 0.2 1 875 equals 0.2 1 8 7 5 which is equal to 21 21 point you know if I were to round roughly twenty one point nine percent chance so there's a little bit better than one in five chance that I get exactly three out of the eight flips as heads hopefully I didn't confuse you and now you know you can apply that top pretty much anything you could say well what is the probability of getting if I flip a fair coin of getting exactly seven out of eight heads or you can say what's the probability of getting you know two out of 100 heads and you could you could you know use the exact same way we did this problem I'll see in the next video