Example: Lottery Probability | Probability And Combinatorics | Precalculus | Khan Academy Video



Winning the Lottery


What is the probability of winning a 4-number lottery?

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To win a particular lottery

game, a player chooses 4numbers from 1 to 60.Each number can only

be chosen once. If all 4 numbers match the 4

winning numbers, regardless oforder, 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’regoing to choose four

numbers from 60.

So one way to think about it

is, how many differentoutcomes are there if we choose

four numbers out of 60?Now this is equivalent to

saying, how many combinationsare there if we have 60 items?In this case we have 60 numbers,

and we are going tochoose four.And we don’t care

about the order. That’s why we’re dealing

with combinations, notpermutations. 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 formulahere, but it’s important to

understand the reasoningbehind the formula. I’ll write the formula here,

but we’ll think about whatit’s actually saying. So this is 60 factorial over 60

minus 4 factorial, dividedalso by 4 factorial,

or the denominatormultiplied 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 just60 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 ofout 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 ofpermutations. You could pick four items out of

60 without replacing them. Now, this is when you cared

about order, but you’reovercounting because it’s

counting differentpermutations that are

essentially the samecombination, 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 waysthat four numbers can be

arranged in four places. Right?The first number can be in one

of four slots, the second inone 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 therefor the lottery game. So this is equal to we already

said the blue part isequivalent to 60 times 59,

times 58, times 57. So that’s literally 60

factorial divided byessentially 56 factorial.

And then you have your 4

factorial over here, which is4 times 3, times 2, times 1.And we could simplify it a

little bit just before webreak 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 ifwe choose four numbers

out of 60 and wedon’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 fournumbers out of 60, or

60 choose four. Now, the question they say is,

what is the probability thatthe winning numbers are

3, 15, 46, and 49?Well, this is just one

particular of thecombinations. This is just one of the 487,635

possible outcomes.

So the probability of 3, 15, 46,

49 winning is just equalto 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 orcombinations when you take 60

and you choose four from that. .

.

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