Probability & combinations (2 of 2)

welcome back I actually recorded this video earlier today but I realized my microphone wasn't plugged in and I won't name names and in terms of who unplugged it but anyway back to probability my wife is giggling mischievously anyway so let's do a slightly harder problem that we did before we were dealing with fair coins let's deal with a slightly unfair coin let's say I have a coin and it's actually said a fun fair coin let's let's do basketball let's say I'm shooting free-throws and I have a free-throw percentage of free-throw percentage is 80% so when I shoot a free-throw eight out of ten times or eighty percent of the time I will make it but that also says that 20% of time I will miss it so given that if I were to take I don't know five free throws what is the probability what is the probability that I make at least three of the five free throws what is the probability let's think of it this way what is the probability of any particular combination of making three out of five so what do I mean by that let's pick let me pick a particular combination let's say it's a basket basket basket and then I miss miss so that would be you know I made three out of the five it could be I don't know basket miss basket miss basket and there's a bunch of them and we'll actually try to figure out how many of them there are but what is the probability of this particular combination well I have an 80% chance of making this first basket times 80% that's the times right there times 80% and then what's my probability of missing well that's 20% right times 0.2 times 0.2 so this equals 0.8 to the third power times 0.2 squared well what's the probability of getting this exact combination well it's 0.8 times then I miss that's there's a 20% chance of that so times 0.2 times 0.8 times 0.2 times 0.8 right we can rearrange this because when you multiply numbers it doesn't matter what order you multiply them in so this is the same thing as 0.8 times 0.8 times 48 times 0.2 times 0.2 so this is also the same thing as 0.8 to the third times 0.2 squared so pretty much any particular the probability of getting any particular combination of three baskets and two misses is going to be 0.8 to the third times 0.2 squared now what's the total probability of getting three out of five well it's going to be the sum of all of these combinations you know I could list them all but we hopefully now are proficient enough in in in combinatorics and combinations to figure out how many different ways if we have five baskets and we're picking or we have five shots and we're picking three of them to be the ones that are basket shot so what do I mean so let's say my my five shots so you know I have shot 1 2 3 4 5 and I'm going to out of these 5 I'm going to choose 3 so I'm once again I'm putting my hat on as the god of probability and I will choose 3 of these shots to be the ones that happen to be the ones that get made so essentially out of 5 out of 5 I am choosing 3 5 choose 3 and what is that equal to that's 5 factorial over 3 factorial times 5 minus 3 factorial so that's 2 factorial and that equals 5 times 4 times 3 times 2 times 1 over 3 times 2 times 1 times 2 times 1 we can ignore all the ones let's see we get 3 times 2 times 1 3 times 2 times 1 we can cancel that this one we can ignore and then this 2 and then this turns into 2 so there are 10 possible combinations these are two of them you know basket basket basket miss miss basket miss basket miss bass and you know that's a it's a good exercise for you to list the other eight of them but using just our you know the binomial coefficient and hopefully you have an intuition of why that works and I'd be happy to make more videos if you feel that you need more explanation whether I made a couple there are ten combinations so essentially the probability of getting exactly three out of five baskets if I am an 80% free-throw shot is going to be switch colors the probability three out of five baskets is going to be equal to the probability of each of the combinations right which is 0.8 to the third 0.8 to the third times 0.2 squared right I make I make three ms2 and then times the total number of combinations right each of these has a probability of this much and then there's ten different arrangements that I could make there's ten different ways of getting three baskets and two misses so times 10 and what does that equal to let me get my my high-end calculator here so let's see what that is that is 0.8 times 0.8 times 0.8 times 0.2 times 0.2 times 10 equals 20 point four eight so it's essentially a twenty point four eight percent chance that I get exactly three out of five of the baskets now let's say if let's make it a slightly more interesting let's say I don't want to the probability of three out of five and then this is actually something that probably people are more likely to ask what is the probability what is the probability of getting at least three at least three baskets well if you think about it this is the probability this is equal to the probability of getting three out of five baskets plus the probability of getting exactly four out of five baskets plus the probability of getting exactly five out of five baskets right we already figured this one out that's twenty point four eight percent so what's the probability of getting four out of five baskets well once again what is four what if we want exactly four out of five baskets so you know an example could be how to know Miss MIT basket basket basket basket what's the probability of any one of the combinations where I make four baskets well it's going to be 0.8 to the fourth times my and then I have a 20% chance of that one miss right and then you know it could have been basket miss basket basket basket right that's the same that's also exactly four but when you multiply them the public of getting any one of these particular combinations is exactly this point eight to the fourth times 0.2 and so how many ways can you if I have five baskets how many ways can I pick four of them to be the ones that I make if I'm once again the god of probability so this this is going to be 0.8 to the fourth times 0.2 times 0.2 times out of five baskets I'm choosing for that I'm going to make so this is the number of combinations where I get four out of the five so what is five choose four that's five factorial over four factorial times one factorial well that equals that equals just five you can work that out so it's going to be so let's just figure this out so it's going to be 0.8 times 0.8 times 0.8 is three times 0.8 it equals no did I do that right let's see point one point eight 0.8 times 0.8 yeah that's right times 0.2 times five times five so forty point nine six percent so this is forty point nine six percent so roughly forty one percent chance that I get exactly four out of five baskets which is interesting because that's kind of my free-throw percentage so it's almost a little less you know two-third shot of kind of hitting my free-throw percentage on the mark on that time and that's probably of getting five out of five well there's only one way of getting five out of five to get it have to get all five of them so this is 0.8 to the fifth power let me get the calculator back so it's 0.8 times 0.08 times 0.8 times 0.8 times 0.8 times 0.8 equals 0.32 7-6 so 0.3 so 32.7 7% shot and then we could add them all up right because we want the probability of at least three so it's going to be that the probability getting 5 out of 5 plus the probably getting of four out of five which is 0.40 nine six plus the probability of getting three out of five so that's 0.20 four eight equals 0.9 for 208 so ninety four point two one roughly rounding percent chance which makes sense if I have an 80% free-throw up a percentage on any one shot I have a very high probability of getting at least three out of five when I go to the free-throw line anyway I'm all out of time I'll see you in the next video