Probability for choosing couples in a party

combinatoricsprobability

there are $N$ couples in a party. $2K$ people from the party are invited to play a game $(2K\leq N)$.
What is the probability that out of the $2K$ people that were invited to play the game:

$\mathbf A.$ exactly $K$ couples were invited to play

$\mathbf B.$ exactly $one$ couple was invited to play
$$$$
this is what I did:

first I calculated the Omega: there is no order (it doesn't matter if one couple was chosen before the other) and after we chose a couple we can't choose them again, so:
$$|\Omega| = \frac{N!}{(N-2K)!}$$

now we can look at a couple as an individual- so I have to chose $K$ couples from $N$ couples, so:

$$P(A) = \frac {|A|}{|\Omega|} = \frac {\frac {N!}{(N-K)!}} {\frac {N!}{(N-2K)!}} = \mathbf{\frac{(N-2K)!}{(N-K)!}}$$

Am I right?

for $\mathbf B$ I am a little confused. I know I need to choose one couple so: ${N \choose 1}$ , but now I'm not sure how to continue.

Thank you!

Best Answer

There are $N$ couples at a party, from which $2K$ people are selected to play a game. What is the probability that out of the $2K$ people invited to play the game, exactly $K$ couples were selected?

The sample space consists of all subsets of $2K$ people that can be selected from the $2N$ people at the party. Hence, $$|\Omega| = \binom{2N}{2K} = \frac{(2N)!}{(2K)!(2N - 2K)!}$$ In your sample space, you counted ordered selections of $2K$ people from the $N$ couples at the party. While you could use ordered selections if you also use ordered selections for the favorable cases, you should have selected $2K$ people from the $2N$ people at the party.

Observe that if exactly $K$ couples are selected, then all $2K$ of the people selected to play the game must be members of those $K$ couples. Hence, the number of favorable cases is the number of ways we can select exactly $K$ of the $N$ couples at the party, which is $$\binom{N}{K}$$ Hence, $$\Pr(\text{exactly $K$ couples are selected}) = \frac{\dbinom{N}{K}}{\dbinom{2N}{2K}}$$

There are $N$ couples at a party, from which $2K$ people are selected to play a game. What is the probability that out of the $2K$ people invited to play the game, exactly one couples was selected?

Strategy: Use the same sample space as above. For the favorable cases:

  1. Select from which of the $N$ couples both members will be invited to play the game.
  2. That leaves you with $2K - 2$ people to select to play the game, each of whom must come from a separate couple. From the remaining $N - 1$ couples, select $2K - 2$ couples from which a representative may be selected (here we use the fact that $2K \leq N$).
  3. Choose a representative from each of those $2K - 2$ couples.
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