Mega millions jackpot probability

I've been asked to calculate the probability of winning the Mega Millions jackpot so I thought that's what I would do this video on so the first thing is make sure we understand what does winning the jackpot actually mean so there's going to be two bins of balls so you're going to have two bins of balls one of them is going to have 56 balls in it so 56 and 1 bin and then another bin is going to have 46 balls in it so there are 46 balls in this bin right over here and so what they're going to do is are going to pick five balls from this bin right over here and you have to get the exact numbers of those five balls it can be in any order so let me just draw them so it's one ball I'll shade it so it looks like a ball two balls 3 balls four balls and five balls so they're going to pick and you just have to get the numbers in any order so this is from a bin of 56 from a bin of 56 and then you have to get the Mega Ball right and then you're going to just pick one ball from there which they call the Mega Ball they're going to pick one ball from there and obviously this is just going to be picked this is going to be one of 46 so from a bin of 46 and so to figure out the probability of winning it's essentially going to be one of all of the possible all of the possibilities of numbers that you might be able to pick so essentially all of the combinations of the white balls times the 46 possibilities that you might get for the Mega Ball so to think about the combinations for the white balls there's a couple of ways you could do it if you are used to thinking in combinatorics terms it would essentially saying well out of a set of 56 things I am going to choose I am going to choose five of them so this is literally you could view this as 56 choose five or if you want to think of it in more conceptual terms the first ball I'd pick there's 56 possibilities since we're not replacing the ball the next ball I pick there's going to be 55 possibilities the ball after that there's going to be 54 possibilities ball after that there's going to be 53 possibilities and then the ball after that there's going to be 52 possibilities 52 because I've already picked four balls out of that now this number right over here when you multiply it out this is the number of permutations if I cared about or so if I got that exact combination but to win this you don't have to you don't have to write them down in the same order it could actually you just have to get those numbers in any order and so what you want to do is you want to divide this by the number of ways the number of ways that five things can actually be ordered so what you want to do is divide this by the way that five things can be ordered and if you're ordering five things the first of the five things can take five different positions then the next one will have four positions left then the one after that will have three positions left one after that will have two positions and then the fifth one it will be completely determined because you've already placed the other four so it's going to have only one position so when we calculate this part right over here this will tell us all of the combinations of just the white balls and so let's calculate that so just the white balls we have 55 sorry 56 times 55 times 54 times 53 times 52 and we're going to divide that by five times four times three times two we'd have to multiply by one but I'll just do that just to show what we're doing and then that gives us about 3.8 million so let me actually take that let me put that off screen so let me write that number down so this comes out - this comes out to three million eight hundred and nineteen thousand eight hundred and sixteen so that's the number of possibilities here so just your odds of picking just the white balls right are going to be one out of this assuming you only have one entry and then there's 46 possibilities for the orange ball so you're going to multiply that times 46 and so that's going to get you so when you multiply it times 46 bring the calculator back so we're going to multiply our previous answer times 46 ants just means my previous answer times 46 I get a little under one hundred and seventy-six million a little under one hundred seventy-six million so that is so that is let me write that number down so that gives us that gives us 175 million seven hundred and eleven thousand and five hundred and thirty six so your odds of winning it what one entry because this is the number of possibilities and you are essentially for a dollar getting one of those possibilities your odds of winning is going to be one over this and to put this in a little bit of context I looked it up on the internet what your odds are of actually getting struck by lightning in your lifetime and so your odds of getting struck by lightning in your lifetime is roughly one in 10,000 one in 10,000 chance of getting struck by lightning in your lifetime and we can roughly say your odds of getting struck by lightning twice in your lifetime or another way of saying it is the odds of you and your best friend both independently being struck by lightning when you're not around each other is going to be 1 in 10,000 times 1 in 10,000 and so that will get you 1 in and we're going to have now 8 zeros 1 2 3 4 5 6 7 8 so that gives you one in a hundred million so you're actually twice almost this is very rough you're roughly twice as likely to get struck by lightning twice in your life than to win the mega jackpot