Mega Millions Jackpot Probability

Probability of winning the Mega Millions jackpot

I've been asked to calculate the probability of winning

The Mega Millions Jackpot

So that's what I thought I'd do this video on.

So the first thing is to make sure we understand what winning the jackpot actually means.

So there are going to be two bins of balls.

One of them is going to have 56 balls. So 56 balls in one bin.

And then another bin will have 45 balls in it.

So 46 balls in this bin right here.

So they are going to pick 5 balls from this bin right here, and you have

to get the exact number of these five balls. It can be in any order, so let me just draw

So its one ball, all shaded so it looks like a ball.

two balls, three balls, four balls

and five balls

And you just have to get the numbers in any order. And this is from the bin of 56

From a bin of 56.

And then you have to get the megaball right.

and they're just going to pick one ball from there, which they call the megaball.

They're going to pick one ball from there. And obviously this is going to be one of 46

So, from a bin of 46.

So to figure out the probability of winning is going to be one of all the possible

all of the possibilites of the numbers you can pick

Its essentially all the combination of the white balls times the 46 possibilites for the mega ball

So, think about the combinations about the white balls

There are couple ways you could do it

if you are thinking in combinatorix terms

it would essentially be like saying

out of a set of 56 things, I am going to choose

I'm going to choose 5 of them

Literally, you could do this as

56 choose 5

Or, if you want to think of it in more conceptual terms

the first ball I pick there is 56 possiblities

Since we're not replacing the ball

The next ball I pick

There are going to be 55 posibilites

the ball after that there is going to be 54 possibilities.

The ball after that there will be 53 possibilities

The ball after that there is going to be 52 possiblites.

52, because I've already picked 4 balls

Now this number right over hre

when you multiply it out is the number of permutations

if I cared about order

If I got that exact combination

But to win this, you don't have to write it down in the same order

You just have to get those numbers in any order

So what you want to do is want to divide this by the number of ways that five things can actually be

ordered.

And if you're ordering five things

The first of the five things could take five different positions

And the next one will have 4 positions left

And the one after that will have 3 positions left

and the one after that will have two positions

And the fifth one will be completely determined because we've already placed the other four.

So it will only have one position

So when we calculate this part right over here

This will tell us all of the combination of just the white balls

So let's calculate that

So just the white balls, we have 55

sorry, 56 x 55 x 54 x 53 x 52

and we're going to divide that

by 5 x 4 x 3 x 2 x 1

but we don't have to mutiply by one, but I'll do that just to show what we're doing

And that gives us about 3.8 million

So let me put that off screen, and let me write that number down

So this comes out to 3 million 8 hundred and 19 thousand

3,819,816

So thats the number of possiblities here

So just your odds of just picking the white balls right

are one out of this

assuming you only have one entry

and then there are 46 possibilities for the orange balls

so you're going to multiply that by 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

Times 46, I get a little under 176 million

A little under 176 million

So that is, let me write that number

So that gives us 175,711,536

So your odds of winning it, with one entry

because this is the number of possiblities

And essentially for $1, you are getting one of those possibilities.

Your odds of winning is going to be one over this

And to put this into a little context

I looked it up on the internet of what your odds are of actually getting struck by lightning in your

lifetime

And your odds of getting struck by lightning are 1 in 10,000

1 in 10,000 chance of getting struck by lightning in your lifetime

And we could roughly say that the odds of getting struck twice in your life time

Or another way of saying it, the odds of you and your best friend

both independently getting struck by lightning when you're not around eachother

Is going to be 1 in 10,000 times 1 in 10,000

and so that will get you 1 in - I have now eight zeros

one, two, three, four, five, six, seven, eight.

So that gives you one in one hundred million

So you 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 MegaJackpot!