Probability using combinatorics
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 to make sure we understand what does winning the jackpot actually mean. So there's going to be two bins of balls. One of them is going to have 56 balls in it, so 56 in one 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 they're going to pick 5 balls from this bin right over here. And you have to get the exact numbers of those 5 balls. It can be in any order. So let me just draw them. So it's 1 ball-- I'll shade it so it looks like a ball-- 2 balls, 3 balls, 4 balls, and 5 balls that they're going to pick. And you just have to get the numbers in any order. So this is from a bin of 56. And then you have to get the mega ball right. And then they're going to just pick one ball from there, which they call the mega ball. 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 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 say, well, out of a set of 56 things, I am going to choose 5 of them. So this is literally, you could view this as 56, choose 5. Or if you want to think of it in more conceptual terms, the first ball I 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, because I've already picked 4 balls out of that. Now, this number right over here, when you multiply it out, this is a number of permutations, if I cared about order. So if I got that exact combination. But to win this, you don't have to write them down in the same order. 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 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, and then the one after that will have three positions left. The one after that will have two positions. And then the fifth one 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 56 times 55 times 54 times 53 times 52. And we're going to divide that by 5 times 4 times 3 times 2. We don't have to multiply by 1, 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 let me put that off screen. So let me write that number down. So this comes out to 3,819,816. 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 balls, 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. "Ans" just means my previous answer. I get a little under 176 million. Let me write that number down. So that gives us 175,711,536. So your odds of winning it, with one entry-- because this is the number of possibilities, and you are essentially, for $1, getting one of those possibilities. Your odds of winning is going to be 1 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 are roughly 1 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 eight 0's-- 1, 2, 3, 4, 5, 6, 7, 8. So that gives you 1 in 100 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.