Probability with permutations & combinations example: taste testing | Probability & combinatorics

• [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.