### Video Transcript

A group of 𝑛 badminton players are trying to pick a captain and two vice captains for their team. They put each of their names into a hat. Three names are selected from the hat at random.

Part a) Jenny’s name is in the hat. Show that the probability that her name does not get picked for any of the three positions is 𝑛 minus three over 𝑛.

There’s also part b that we’ll come to later. So if we’re looking to solve this problem, first, we will need some notation. So we’re gonna use 𝐽 prime to mean “not Jenny.”

So if we think about what’s gonna happen for each pick, well, with our first pick, the probability that it’s not going to be Jenny — and again we’ve got 𝑃 here using some probability notation and then in brackets 𝐽 prime — is going to be equal to 𝑛 minus one. That’s because 𝑛 is the number of players.

And then there’s only one Jenny, so we take away one. So that means everyone else is not Jenny. So that’s why we’ve got 𝑛 minus one as our numerator. And then divided by 𝑛 because that’s our total number of possible outcomes, because that’s the total number of players.

Okay, so that’s for the first pick. Well, if you think about the second pick, then the probability of it not being Jenny for the second pick is going to be equal to this time 𝑛 minus two as the numerator. And that’s because we’ve got 𝑛 is the total number of players, then minus one, because, all we see, we’re not picking Jenny.

But also we’ve already picked one person from the hat in the first round of picks. So that’s why it’s minus two. And this is gonna be divided by this time 𝑛 minus one. And that’s because 𝑛 was the total number of players to begin with. But we’ve already picked one from the hat already. So now we’ve got 𝑛 minus one left.

And then if we think about the third pick, well, the probably of it not being Jenny this time is gonna be equal to, well, in the numerator, we’re gonna have 𝑛 minus three, because we’ve got Jenny and the two people have already been picked, over 𝑛 minus two, because that’s 𝑛, the number of players we started with, minus two, because that’s the two picks we’ve already had.

So now we’ve got the probabilities for the first, second, and third picks that it’s not gonna be Jenny that’s picked. Well, to solve the problem, what we need to do is show that the probability that her name does not get picked for any of the three positions is 𝑛 minus three over 𝑛. So how do we use the information we’ve got?

Well, if we think about it using a tree diagram, it would look a bit like this. So we’ve got our branches. So we can say that we’ve got not Jenny and Jenny for the first pick, not Jenny and Jenny for the second pick, and then not Jenny and Jenny for the third pick.

It’s interesting to note that, each of the times, the probability of picking Jenny is gonna be one out of however many players there are left, so one out of 𝑛 or one out of 𝑛 minus one or one over 𝑛 minus two. Well, if we want to know the probability that she does not get picked for any of the three positions, this is the same as the probability of not Jenny, not Jenny, not Jenny.

And remembering how we use our tree diagrams, we’re actually gonna multiply each of the probabilities along the branches. So we’ve got 𝑛 minus one over 𝑛 multiplied by 𝑛 minus two over 𝑛 minus one multiplied by 𝑛 minus three over 𝑛 minus two.

Well, this looks quite tricky. But what we can do is we can actually cross-cancel to help ourselves out. So first of all, our 𝑛 minus ones can cancel, because we’ve got 𝑛 minus one on numerator and the denominator. And then our 𝑛 minus twos can cancel, because again we’ve got one on the numerator and one on the denominator. So therefore, we can say that the probability that Jenny does not get picked for any of the three positions is gonna be equal to 𝑛 minus three over 𝑛.

So now to part b, part b) says the probability that Jenny’s name does not get picked is greater than 0.8. We need to calculate the smallest possible number 𝑛 of people in the team.

So therefore, to calculate this and work out the smallest possible number 𝑛 of people in the team, we need to set up an inequality. And we’ve got one here because we know that the probability that Jenny’s name does not get picked is greater than 0.8. So therefore, we’ve got 𝑛 minus three over 𝑛 is greater than 0.8. And we know to put the inequality sign this way round because the thing that is greater is always at the open end of the inequality.

So now to solve the inequality, what we’re gonna do is multiply each side of the inequality by 𝑛. So we get 𝑛 minus three is greater than 0.8𝑛. And then next, what we’re gonna do is subtract 0.8𝑛 from each side of the inequality and add three to each side of the inequality. And when we do that, we get 0.2𝑛 is greater than three. And we get that because 𝑛 or one 𝑛 minus 0.8 is gonna leave us with 0.2. So that’s 0.2𝑛. And we’ve added three to each side of the inequality. So we’ve got three on the right hand of the said inequality.

So now what we want to do to find 𝑛 is we need to divide each side of the inequality by 0.2. So we get 𝑛 is greater than three over 0.2. And this looks quite tricky to do. So let’s make this a bit easier by multiplying the numerator and the denominator by 10.

And when we do this, we’re gonna get 𝑛 is greater than 30 over two. So therefore, we know that 𝑛 is greater than 15. So is this the question solved?

Well, no, because what we want is to calculate the smallest possible number 𝑛 of people in the team. So we’ve got our inequality. But what is gonna be the smallest possible number of 𝑛 people in the team? Well, therefore, 𝑛 must be equal to 16 because we’re told that 𝑛 is greater than 15 but not equal to. So that means 𝑛 can’t be 15. So therefore, the smallest possible number it can be is 16.