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Having a collection of unique items with index $i = {1 .. n}$, a set size $s >= 0$, and total population of all items with size $N = n * s$ (so population has $s$ copies of every unique item).

I select a sample of $m$ random items from total population, $0 <= m <= N$.

Now I select a random index $0 < r <= n$ (any possible item index with even chance).

Assuming I don't know the exact composition of the sample (i.e how many copies of each item I already have) I’d like to know the probabilities that I already have $0, 1, … s$ copies of item with index $r$ in my sample.

Example: In a standard deck of cards there are 13 cards with 4 copies of each (with different colors). If I randomly select a sample of $m$ cards from full deck of 52 cards, what are the chances that I have 0, 1, 2, 3 or 4 copies of any card (e.g. an Ace) in my sample already.

Can anyone help with a formula or algorithm? If possible, a short explanation to help me tackle similar problems.

Does it matter if I know the exact composition of the sample? Do the probabilities change?

Thanks.

(EDIT: added example)

danielv
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1 Answers1

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  • $0$: we need to pick everything but the Aces, so $\binom{48}{m}$ possible chances over $\binom{52}{m}$ ;
  • $1$: we need to pick one Ace, so $4\binom{48}{m-1}$ possible chances over $\binom{52}{m}$ ;
  • $2$: we need to pick two Aces, so $\binom{4}{2}\binom{48}{m-2}$ possible chances over $\binom{52}{m}$ ;
  • $3$: we need not to pick one Ace, so $4\binom{48}{m-3}$ possible chances over $\binom{52}{m}$ ;
  • $4$: we need all the Aces, so $\binom{48}{m-4}$ possible chances over $\binom{52}{m}$.
thomasb
  • 570
  • Thanks. Can you please give some explanation please? It seems to be correct but I'd like to understand exactly why. – danielv Sep 30 '19 at 10:46
  • If we want to pick one Ace only, we have four choices. For each of these choises, we then need to pick $m-1$ cards among the no-Aces which are $48$ cards. The number of possibilities picking any $m$ cards is $\binom{52}{m}$. – thomasb Oct 01 '19 at 10:10