The odds your key card is in the pile you set aside
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Nothing here is tied to one game: any game that puts cards aside unseen before play, where you only find out afterwards what was in there, works out the same way.
Results
| Chance at least one copy is set aside, in per cent | — |
| Chance none of them is, in per cent | — |
| Chance every copy is in there, in per cent | — |
| Copies you can still expect to have available | — |
| What the pile takes from every copy, in per cent | — |
How many of them end up in there
| Copies set aside | Chance of exactly that, in per cent | Chance of that many or more, in per cent |
|---|
The two ends of that table are the whole misunderstanding. Losing one copy is common enough to plan around, and losing all of them is so rare that building a deck to survive it costs more than it saves. The middle of the table is where the honest worry lives.
The steadiest number is the last one in the panel above, because it does not move with how many copies you run. Setting six cards aside from sixty takes a tenth of every copy of every card in the deck, whether you play one of something or four, so the pile is a flat tax rather than a gamble aimed at your best card.
That is why running one more copy helps in the way it does: it does not lower the rate at all, it just leaves you more to lose from. What the page cannot tell you is what the card is worth when you do have it, and that is the half of the decision that decides whether the tax is worth paying.
How often does a card I run four of end up in there?
More often than the headline rate suggests. Setting six cards aside from sixty puts at least one of your four copies in the pile in over a third of games.
The disaster people fear is the opposite: all four in there at once happens in about three games in a hundred thousand, so building around that possibility costs far more than it saves.
| Copies you run | At least one set aside, in per cent | All of them set aside, in per cent |
|---|---|---|
| 1 | 10,0000 | 10,0000 |
| 2 | 19,1525 | 0,8475 |
| 3 | 27,5161 | 0,0584 |
| 4 | 35,1460 | 0,0031 |
Does running more copies make the pile less of a problem?
Not as a rate, which is the part that surprises people. The pile takes the same fraction of every copy of every card in the deck, six from sixty being a flat tenth, whether you run one of something or four.
What more copies buy you is a bigger number left over rather than a smaller share taken: four copies leave three and six tenths available on average, one copy leaves nine tenths.
So is it worth building a deck to get around it?
Against losing one copy, usually not: that is the ordinary case and the cost of insuring against it is paid every game. Against losing every copy, definitely not, because it almost never happens.
The middle of the distribution is where an honest worry lives, and the table shows exactly how much of it there is.
What does the page assume?
That the cards set aside are chosen at random from the whole deck and stay unknown, which is what makes the count a plain hypergeometric one.
Anything that lets you look at the pile, swap cards into it, or set it aside after a search sits outside the model.
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