Main content

### Course: High school statistics > Unit 6

Lesson 7: Probability using combinatorics- Probability using combinations
- Example: Lottery probability
- Example: Different ways to pick officers
- Probability with permutations & combinations example: taste testing
- Probability with combinations example: choosing groups
- Probability with combinations example: choosing cards
- Probability with permutations and combinations
- Mega millions jackpot probability

© 2024 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Probability with permutations & combinations example: taste testing

We can use combinations (when order does not matter) and permutations (when order does matter) to find probabilities. Created by Sal Khan.

## Want to join the conversation?

- who like to drink olive oil?(7 votes)
- why is it a one on top if they pick three different types?(3 votes)
- Shouldn't it be 1/15C3 * 1/15C3? Because not only are you calculating the probability that the participant guesses some 3 specific flavors; but also that those are the 3 selected by Samara.(2 votes)
- I thought that permutations meant that ABC is not equal to BAC, meaning that we don't care about the order, so they both add to the list of permutations individually. But am I interpreting the way we use the word "care" in this case wrongly?

But there would be more outcomes if we didn't care about the order, as instead of just having any one-order-of-letters, ABC, for example, replacing ACB, BAC, BCA, CAB, and CBA, meaning that we would always have more outcomes when using permutations.

In this case, we wouldn't be trying to guess in the right order the three different olive oils, so long as we choose the right three types. Wouldn't that mean that we need to deal with permutations to solve this problem?

But then, at2:00, Sal stated that trying to use permutations would mean that we'd have to guess the correct order. But permutations means that ABC, CAB and BCA, etc. are are different, indicidual permutations, right? Meaning that we wouldn't be needing to guess the correct order, which the question states we don't need to do, guess the correct order of olive oils mixed together.(2 votes) - This is my second time through this lesson (first in statistics, now precalculus) and I still can't quite explain to myself why we do - let's say the combination of 5 items in 3 slots = so 5 C 3, which means that it's 5x4x3 divided by 3x2x1. I mean I understand it enough to solve, but I'm not satisfied, missing some extra spark of insight.(1 vote)
- So you seem to understand the content. You found nPr which is 5*4*3. You then took into account the number of positions. You did this by dividing by 3*2*1 or in other words 3!.

To put it another way nCr = n!/[(n-r)!*r!], where n is the number of object. r is the number of positions available.

The key idea to derive this formula is the rule of product axiom (refer to Wikipedia if need be). An axiom is a statement that is true and cannot be proven.(2 votes)

- And how is it that 1/15*14*13/3*2*1 = 3*2*1/15*14*13? I don't exactly understand that, either.(1 vote)

## Video transcript

- [Instructor] We're told
that Samara is setting up an olive tasting
competition for a festival. From 15 distinct varieties, Samara will choose three
different olive oils and blend them together. A contestant will taste the blend and try to identify which
three of the 15 varieties were used to make it. Assume that a contestant
can't taste any difference and is randomly guessing. What is the probability that a contestant correctly guesses which three varieties were used? So pause this video and see
if you can think about that. And if you can just come
up with the expression, you don't have to compute it. That is probably good enough,
at least for our purposes. All right, now let's work
through this together. So we know several things here. We have 15 distinct varieties and we are choosing
three of those varieties. And anytime we're talking about probability and combinatorics, it's always interesting to
say, "Does order matter? Does it matter what order that Samara is picking those three from the 15?" It doesn't look like it matters. It looks like we just have to think about what three they are. It doesn't matter what order
either she picked them in, or the order in which the
contestant guesses them in. And so if you think about the total number of ways of picking three
things from a group of 15, you could write that as 15, choose three. Once again, this is
just shorthand notation for how many combinations are there, so you can pick three
things from a group of 15? So some of you might
have been tempted to say, "Hey, let me think
about permutations here. And I have 15 things. And from that, I wanna
figure out how many ways can I pick three things that
actually has order mattering?" But this would be the
situation where we're talking about the contestant actually
having to maybe guess in the same order in which the varieties were originally blended,
or something like that, but we're not doing that, we just care about getting
the right three varieties. So this will tell us
the total number of ways that you can pick three out of 15. And so what's the probability that the contestant correctly guesses which three varieties were used? Well, the contestant is
going to be guessing one out of the possible
number of scenarios here. So the probability would be
one over 15, choose three. And if you wanted to compute this, this would be equal to one over, now, how many ways can you
pick three things from 15? And of course there is a formula here, but I always like to reason through it. Well, you could say, "All
right, if there's three slots, there's 15 different varieties that could've gone into that first slot, and then there's 14 that could
go into that second slot, and then there's 13 that can
go into that third slot." But then we have to remember that it doesn't matter
what order we pick them in. So how many ways can you
rearrange three things? Well, it would be three factorial, or three times two times one. So this would be the same thing
as three times two times one over 15 times 14 times 13. See, I can simplify this, divide numerator and denominator by two, divide numerator and denominator by three. This is going to be equal
to one over 35 times 13. This is going to be one over
350 plus 105, which is 455. And we are done.