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## Probability using combinatorics

# Example: Lottery probability

CCSS.Math:

## Video transcript

To win a particular lottery
game, a player chooses 4 numbers from 1 to 60. Each number can only
be chosen once. If all 4 numbers match the 4
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 four
numbers from 60. So one way to think about it
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 four. 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
four can we pick out of 60? And we don't care what order
we picked 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 formula. I'll write the formula here,
but we'll think about what it's actually saying. So this is 60 factorial over 60
minus 4 factorial, divided also by 4 factorial,
or the denominator multiplied by 4 factorial. So this is the formula
right here. But what this is really saying,
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 1 of 60 numbers, but
then that number is kind of out of the game. Then you can pick from 1 of
59, then from 1 of 58, then of 1 of 57. So if you cared about order,
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 overcounting because it's
counting different permutations that are
essentially the same combination, essentially the
same set of four numbers. And that's why we're dividing
by 4 factorial here. Because 4 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 in one of three, then
two, then one. That's why you're dividing
by 4 factorial. But anyway, let's just
evaluate this. This'll tell us how many
possible outcomes are there for the lottery game. So this is equal to-- we already
said the blue part is equivalent to 60 times 59,
times 58, times 57. 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. And let's see, we have a
58 divided by 2 is 29. So our answer is going to be 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 four 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. It's equal to 487,635. So let me write that down. That is 487,635 combinations. If you're picking four numbers,
you're choosing four numbers out of 60, or
60 choose four. Now, the question they say is,
what is the probability that the winning numbers are
3, 15, 46, and 49? Well, this is just one
particular of the combinations. This is just one of the 487,635
possible outcomes. So the probability of 3, 15, 46,
49 winning is just equal to-- well, this is just one of
the outcomes out of 487,635. So that right there is your
probability of winning. This is one outcome out of all
the potential outcomes or combinations when you take 60
and you choose four from that.