Probability with combinations example: choosing groups | Probability & combinatorics

We're told that Kyra works on a team of 13 total people. Her manager is randomly selecting three members from her team to represent the company at a conference. What is the probability that Kyra is chosen for the conference? Pause this video and see if you can have a go at this before we work through this together.

All right, now let's work through this together. So we want to figure out this probability, and so one way to think about it is: what are the number of ways that Kyra can be on a team or the number of possible teams with Kyra, and then over the total number of possible teams—total number of possible teams?

And if this little hint gets you even more inspired, if you weren't able to do it the first time, I encourage you to try to pause it again and then work through it.

All right, now I will continue. So first, let me do the denominator here. What are the total possible number of teams? Some of y'all might have found that a little bit easier to figure out. Well, we know that we're choosing from 13 people and we're picking three of them, and we don't care about order. It's not like we're saying someone's going to be president of the team, someone's going to be vice president, and someone's going to be treasurer. We just say there are three people in the team.

And so this is a situation where out of 13, we are choosing three people. Now, what are the total number of teams possible that could have Kyra in it? Well, one way to think about it is if we know that Kyra's on a team, then the possibilities are: who's going to be the other two people on the team? And who are the possible candidates for the other two people? Well, if Kyra is already on the team, then there's a possible 12 people to pick from. So there's 12 people to choose from for those other two slots.

And so we're going to choose two, and once again we don't care about the order with which we are choosing them. So once again, it is going to be a combination. Then we can just go ahead and calculate each of these combinations here. What is 12 choose 2? Well, there's 12 possible people for that first non-Kyra seat, and then there would be 11 people there for that other non-Kyra spot.

And of course, it's a combination; we don't care what order we picked it in. And so there are two ways to get these two people. We could say two factorial, but that's just the same thing as two or two times one. And then the denominator here for that first spot—there's 13 people to pick from. Then in that second spot, there are 12, then in that third spot, there are 11. And then, once again, we don't care about order—three factorial ways to arrange three people.

So I could write 3 times 2, and for kicks, I could write 1 right over here. And then we can, let's go down here. This is going to be equal to my numerator over here. It's going to be 6 times 11, and then my denominator is going to be 12 divided by 6. Right over here is 2. So it's going to be 13 times 11 times 2.

Just to be clear, I divided both the denominator and this numerator over here by 6 to get 2 right over there. Now this cancels with that, and then if we divide the numerator and denominator by 2, this is going to be 3 here; this is going to be 1. And so we are left with a probability of 3/13 that Kyra is chosen for the conference.