Chance of opening one from each group
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The groups have to be different cards from one another. A card that belongs to two of them at once breaks the count, because it would be filling both jobs with one slot.
Results
Chance of opening at least one from every group
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| What you get by multiplying the groups as if they were independent | — |
| How much that overstates it | — |
| Chance of at least one from the first group on its own | — |
| The same for the second group | — |
| Chance of at least one from any of them | — |
What happens as the second group grows
| Cards in the second group | Real chance | What multiplying would say |
|---|
The gap between the two columns is widest when both groups are middling and both are fighting hard for the same few slots. Checked across every pair of group sizes that fits in a forty card deck with a five card hand: multiplying never once came out below the true figure, and at its worst it was three points high.
For the chance of opening something from any of the groups rather than one from each, adding the separate chances together is worse still. With twelve cards in one group and ten in the other it comes to more than one and a half, which is not a probability at all, and the honest figure is the one in the table.
This page counts groups, not copies. If what you want to know is the chance of drawing two or three of one particular card, that is a different question with its own page, and this one would answer it badly.
Why not just multiply the two chances together?
Because the groups are drawn from the same deck and compete for the same slots in your hand. Every card of one group that turns up is a slot the other one cannot use, so the two events pull against each other rather than being independent.
Multiplying therefore comes out high every time. With twelve cards in one group and ten in the other, in a forty card deck with a five card hand, the real figure is 64.71 percent and multiplying gives 66.64. Checked across every pair of group sizes that fits: multiplying never once landed below the truth, and at its worst it was three points high.
When is the error biggest?
When both groups are middling. Two groups of seven in a forty card deck give the widest gap, because that is where they are fighting hardest over the same five slots without either being nearly certain on its own.
When a group is very large the gap shrinks, since a card that is almost guaranteed to appear stops competing in any meaningful sense. That is why the mistake tends to survive: it looks harmless in exactly the cases people check it on.
What about the chance of opening any of the groups rather than one of each?
That is a different figure, and the page shows it too. Adding the separate chances together is the wrong move there for a related reason: with twelve and ten cards the sum comes to more than one and a half, which is not a probability at all.
The right answer is to treat the groups as one bigger pile and ask for the chance of missing all of it, which is what the table gives.
Can a card belong to two groups at once?
Not in this count. A card that does both jobs would be filling two roles with a single slot, and the arithmetic here assumes each card belongs to at most one group.
If you have cards like that, the honest thing is to count them in whichever group you actually need them for and be aware that the result is then a floor rather than the exact figure.
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