Example: Lottery probability

to win a particularly game a player chooses four numbers from 1 to 60 each number can only be chosen once if all four numbers match the four winning numbers regardless of order the player wins what is the probability that the winning numbers are 3 15 46 and 49 so the way to think about this problem they say that we're going to choose 4 numbers from 60 so one way to think about is how many different outcomes are there if we choose four numbers out of 60 now this is equivalent to saying how many combinations are there if we have 60 items in this case we have 60 numbers and we are going to choose 4 and we're going to choose 4 and we don't care about the order that's why we're dealing with combinations not permutations we don't care about the order so how many different groups of 4 can we pick out of 60 and we don't care what order we pick them in and we've seen in previous videos that there is a formula here but it's important to understand the reasoning behind the form I'll write the formula here but it will think about what what it's actually saying so this is 60 factorial over 60 minus 4 factorial divided also by 4 factorial in the or in the denominator multiplied by 4 factorial so this is the formula right here but what this is really saying this part right here this part right here 60 factorial divided by 60 minus 4 factorial that's just 60 times 59 times 58 times 57 that's what this expression right here is and if you think about it the first number you pick there's one of 60 numbers but then you that number is kind of out of the game then you can pick from one a 59 then from one a 58 then of one of 57 so if you cared about order this is the number of calm this is the number of permutations you could pick four items out of 60 without replacing them now this is when you cared about order but you're kind of over counting because it's counting different permutations that are essentially the same combination essential same set of four numbers and that's why we're dividing by four factorial here because four factorial is essentially the number of ways that four numbers can be arranged in four places right the first number can be in one of four slots the second one of three then two then one that's why you're dividing by 4 factorial but anyway let's just evaluate this this will tell us how many possible outcomes are there for the lottery game so this is equal to we already said the blue part is the blue part is equivalent to 60 times 59 times 58 times 57 and then so that's literally 60 factorial divided by essentially 56 factorial and then you have your 4 factorial over here which is 4 times 3 times 2 times 1 and we could simplify it a little bit just before we break out the calculator 60 divided by 4 is 15 and then let's see 15 divided by 3 is 5 15 divided by 3 is 5 and let's see we have a 58 divided by 2 58 divided by 2 is 29 so the our answer is going to be 5 5 times 59 times 29 times 57 now this isn't going to be our answer this is going to be the number of combinations we can get if we choose 4 numbers out of 60 and we don't care about order so let's take the calculator out now so we have 5 times 59 times 29 times 57 is equal to four hundred and eighty-seven thousand six hundred and thirty-five so let me write that down let me write that down that is that is four hundred eighty seven thousand six hundred and thirty-five combinations combinations if you picking four numbers you're choosing four numbers out of 60 or 60 choose four now with the question they says what is the probability that the winning numbers are three 46 and 49 well this is just one particular of the combinations this is just one of thee this is just one of the four hundred eighty-seven thousand six hundred thirty-five possible outcomes so the probability the probability of three fifteen forty six forty nine winning winning is just equal to well this is just one of the outcomes out of four hundred eighty-seven thousand six hundred and thirty-five so that right there is your probability of winning this is one outcome out of all of the potential outcomes or combinations when you take sixty and you choose four from that