Suppose that three of the labelled points are chosen at random. What is the probability
that the chosen points are collinear?
Well, first let’s just check what collinear
means. It means that the points lie in a straight line. So we’ve got five points to
choose from. We’re gonna choose three of them at random. What’s the probability that
those three points all lie on a straight line?
Now looking at those five points, the only three which all lie on the same straight
line are 𝐹, 𝐺, and 𝐻. Now it doesn’t matter whether we pick 𝐻 first then 𝐺 then
𝐹 or 𝐹 then 𝐺 then 𝐻 or any other order of those three points, but they’re the
three points that we must pick in order to get collinear points.
So we’ve got five different points to choose from, and we need to choose three of them.
So we’re gonna use our 𝑛-choose-𝑟 formula where 𝑛, the number of different points,
is five and 𝑟, the number that we need to choose, is three.
Now there are a lot of different ways of notating that particular formula, so you
choose the one that’s familiar to you. But they all boil down to this calculation: 𝑛
factorial over 𝑟 factorial times 𝑛 minus 𝑟 factorial. So if we just plug our
numbers in, 𝑛 is five and 𝑟 is three, we get five factorial over three factorial
times five minus three factorial. Well five minus three is two.
So this simplifies to five factorial over three factorial two factorial. And of course
five factorial means five times four times three times two times one; three factorial
means three times two times one; and two factorial is just two times one. Now we can
do a bit of cancelling The threes, the twos, and the ones there cancel. So I’ve got
five times four over two times one, which is 20 over two, which is 10.
So there are 10 ways of choosing three different points from our five different points.
But only one of those ways involves having 𝐻, 𝐺, and 𝐹 as the three letters that
we’re interested in.
So out of the 10 different ways that we’ve got of selecting three letters from five
only, one of them is that the group 𝐹, 𝐺, 𝐻. So that means only one out of the 10
possible outcomes results in the chosen points being collinear, so the answer is the
probability that they are collinear is one over 10, a tenth.