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Probability with combinations example: choosing groups

We can use two combinations (when order does not matter) to find the probability of someone being included in the group that's chosen at random from a larger group. Created by Sal Khan.

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  • old spice man green style avatar for user Cal
    Is it just a coincidence that there were 13 members on the team and 3 people were selected per team and the odds of any individual being selected was 3/13? Or is there some tantalizing shortcut that we haven't been introduced to yet?
    (6 votes)
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    • duskpin ultimate style avatar for user Alia
      It's not a coincidence. If one person out of 13 was being picked at random, then Kyra would have a 1/13 chance of being picked. Since the manager is picking three people, Kyra's chance of getting picked is tripled, so P(Kyra getting picked) = 3*(1/13) = 3/13
      (4 votes)
  • blobby green style avatar for user bobbysundstrom
    I'm not sure why this doesn't make sense to me, but I don't understand what Sal means when he says that "if we know that Kyra's on the team, the possibilities are the other two people on the team." Why do the other two people matter?
    (4 votes)
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    • mr pink green style avatar for user David Severin
      One slight correction, Sal says, "if we know that Kyra's on the team, then the possibilities are who's gonna be the other two people on the team, and who are the possible candidates for the other two people?" Adding the who's (even though transcript uses whose) turns it into a question rather than a statement as you have it. So if Kyra is one, what possibilities do we have for the other two? is how I would paraphrase what Sal is saying.
      (5 votes)
  • blobby green style avatar for user Anthony
    I'm with bobbysundstrom in not getting the reason "why the other two people matter?"
    (3 votes)
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  • leaf yellow style avatar for user louise
    i get 13 choose 3 and 12 choose 2 but why did we not do 11 choose 1?
    (2 votes)
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    • mr pink green style avatar for user David Severin
      Where would this go? The chance that Kyra was on the committee (which is the question) is number of committees Kyra is on/total possible number of committees. 12 choose two tells number of committees that Kyra is on, thus the numerator, and total committees is 13 choose 3 which is the denominator.
      (1 vote)
  • blobby green style avatar for user ProPoop
    I'm confused at . He says something similar to "if we know that Kyra is on a team, then what are the possibilities for the other two people?" And for that, he uses 12 choose 2.
    But wouldn't that be the same thing as saying "if we know that <someone else> is on a team, then what are the possibilities for the other two people?" Why does this only work for Kyra's possibilities?
    (1 vote)
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    • cacteye blue style avatar for user Jerry Nilsson
      It doesn't only work for Kyra.

      If we for example knew that Charlie was on the team, then there are 12C2 possibilities for the other two members.

      However, this doesn't mean that there are 12C2 + 12C2 combinations where either Kyra or Charlie are on the team, because this double-counts the combinations where Kyra and Charlie are both on the team and we would also have to subtract 1C11.
      (2 votes)

Video transcript

- [Instructor] 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 wanna 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, 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 to continue. So first let me do the denominator here. What are the total possible number of teams? Some of y'all might've 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 gonna be president of the team, someone's gonna be vice-president and someone's gonna 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 teams that could have Kyra in it? Well, one way to think about it is if we know that Kyra is on a team, then the possibilities are who's gonna 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 gonna choose two. And once again, we don't care about the order with which we are choosing them. So once again, it is gonna be a combination. And then we can just go ahead and calculate each of these combinations here. What is 12 choose two? Well, there's 12 possible people for that first nine Kyra's seat. And then there would be 11 people there for that other non Kyra's 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 three times two, and for kicks I could write one right over here, and then we can, let's go down here. This is gonna be equal to my numerator over here is gonna be six times 11. And then my denominator is going to be 12 divided by six right over here is two. So it's gonna be 13 times 11 times two. Just to be clear, I divided both the denominator and this numerator over here by six to get two right over there. Now this cancels with that. And then if we divide the numerator and denominator by two, this is gonna be three here. This is gonna be one. And so we are left with a probability of 3/13 that Kyra is chosen for the conference.