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## Probability using combinatorics

Current time:0:00Total duration:5:30

# Mega millions jackpot probability

## Video transcript

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.