How often you have to redraw your opening hand
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Redrawing puts the hand back and starts the draw over, so this is not one question but the same question asked again and again. Nothing here is tied to one game: any opening you have to redo until it holds something works out this way.
Results
| Chance you have to redraw, in per cent | — |
| Redraws per game, on average | — |
| Chance of two or more in a row, in per cent | — |
| What one more card of that kind takes off, in points | — |
| Cards of that kind needed to get under one in ten | — |
What each extra card of that kind buys
| Cards of that kind | Chance of redrawing, in per cent | Redraws per game | Points that card took off |
|---|
The last column falls away steadily, and that is the whole of the deckbuilding advice. Early cards of that kind are worth several points each, later ones barely a point, and there is no cliff anywhere to aim at: you are buying a smaller and smaller improvement at a constant price in slots.
The middle column is the one worth carrying around, because it counts something the single chance hides. Redrawing sends the hand back and starts again, so the bad case is not one redraw but a run of them, and a chance that sounds survivable on its own turns into a run more often than people expect.
What none of this measures is what those cards do once they are in play. A deck can be built to open perfectly and do nothing afterwards, and the arithmetic here would be perfectly happy with it, which is why the count is a floor to clear rather than a thing to maximise.
Is redrawing really that common?
More common than it feels. With sixty cards, a hand of seven and eight cards you must open with, you redraw in about thirty-five games out of a hundred.
And because redrawing starts the draw over rather than fixing it, twelve games out of a hundred need at least two in a row.
| Cards of that kind | Chance of redrawing, in per cent | Redraws per game | Two or more in a row, in per cent |
|---|---|---|---|
| 8 | 34,6406 | 0,5300 | 11,9997 |
| 12 | 19,0647 | 0,2356 | 3,6346 |
| 16 | 9,9223 | 0,1102 | 0,9845 |
How many should I run, then?
There is no cliff to aim at, which is the useful part of the answer. Each extra card of that kind takes off less than the one before: about seven and a half points at first, then five, then four, and under one point by the time you are past twenty.
Getting the chance under one in ten takes sixteen of them in the example above, and every card after that buys very little.
Why does the page count redraws instead of just the chance?
Because the hand goes back and the draw starts again, so the thing that hurts is a run rather than a single miss. A chance that sounds survivable on its own becomes a run more often than people expect.
The average number of redraws is the chance divided by one minus the chance, which is the same shape as any wait for a first success.
What does this not tell me?
What those cards do once they are down. A deck can be built to open perfectly and do nothing afterwards, and this arithmetic would be perfectly happy with it.
Treat the count as a floor to clear rather than a number to maximise, and spend the slots you save on the rest of the deck.
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