4
$\begingroup$

In a group of six people $p_1,\ldots,p_6$, two people are chosen to win a prize (=holiday on Tahiti). List all pairs we can make. This is a sample space $S$.

Is the answer

$$S=\{\{p_i,p_j\}\},1\leq i

or

$$S=\{(p_i,p_j)\},1\leq i,j\leq 6,i\ne j$$?

  • 0
    Does picking $(p_1,p_2)$ generate a different result than picking $(p_2,p_1)$?2012-11-01
  • 0
    If it are "pairs" i guess yes.2012-11-01
  • 2
    No, I think not. In this context (both win same prize), if, say person $p_1 \;\text{and } p_2$ are both chosen, that's no different that saying $p_2 \;\text{and } p_1$ are both chosen, i.e., the pair $(p_1, p_2)$ is no different than the pair $(p_2, p_1)$. If the first person chosen were to win a grand prize, and the second were to win a lesser/different prize, then order would matter, and you'd want to list both pairs.2012-11-01
  • 0
    So if I pick $(p_1,p_2)$ as a pair to win a prize, the result is different than if I pick $(p_2,p_1)$ to win a prize? If I give \$10 each to Joe and Jane, is that different than if I give \$10 to each of Jane and Joe?2012-11-01
  • 0
    In this context both win the same prize.2012-11-01
  • 1
    Then, as I mention in my comment above, since order doesn't matter, you need only list $\{p_1, p_2\}$. Can you solve the problem with that issue cleared up? Think of two slots (one slot for each winner): How many possibilities are there for the first slot? When that first slot is filled, then how many possibilities remain for the second slot?2012-11-01
  • 0
    The problem asked me to write all pairs and asked whether the order matter. It also said that "This is a sample space".2012-11-01
  • 0
    Have you figured out how many "pairs" your list should include? Feel free to edit your question, adding the list you come up with, if you want or need help, or have any reservations about the correctness of your solution.2012-11-01
  • 0
    @amWhy I edited.2012-11-01
  • 0
    Your first answer is correct (see edit by Austin, below). The second answer would allow repetition, in that, e.g., {5, 6} and {6, 5} would both be valid, and we've concluded that order doesn't matter...Now, if you want to list them all (if required in homework), make sure your list consists of 15 distinct pairs.2012-11-01

1 Answers 1

1

The word "pair" is generally used to denote an unordered pair (i.e. a set of size two). Such a pair is usually written $\{p_1, p_2\}$ to emphasize that it is a set (order does not matter). The context here suggests that this is the intended meaning of "pair", since both chosen people appear to win the same prize.

If the problem were such that two different prizes were being awarded, then we would speak of "ordered pairs". Such ordered pairs are usually written $(p_1,p_2)$, which is different from $(p_2,p_1)$, perhaps with the convention that the big prize winner goes in the first slot, while the small prize winner goes in the second.


In terms of sets, you might write your answer as $$ \{\{p_i, p_j\} \mid 1 \leq i < j \leq 6\}. $$ Specifying $i < j$ in the makes certain that you do not list sets like $\{p_1, p_1\}$ (the same person winning twice) or $\{p_2, p_1\}$ (which should have already been counted as $\{p_1, p_2\}$).

It might also be worth noting that there are $\binom{6}{2} = 15$ such pairs, since forming the list is equivalent to finding all the ways to choose two indices out of six in which order does not matter.