Probability tools

Five calculators live at /tools. They answer the questions deck building keeps asking: how often you have drawn what you need by a given turn, what an opening hand actually looks like, and whether a mana base makes its land drops. Each runs in your browser. Nothing you type into them is sent anywhere.

The same maths runs on a deck's analysis tab, where the groups come from your own categories instead of a number you type.

The three tools

All three are hypergeometric: they count how many of the possible ways to deal your cards contain what you asked for, and divide. That is exact, not a simulation, so the same inputs always give the same figure.

Why you cannot multiply the pieces

Three of the tools answer a question about one group of cards. The combo tool answers a question about several at once, and the reason it exists is that the obvious shortcut is wrong.

Given three pieces at 80% each, multiplying gives 51%. That figure assumes the three events are independent. They are not: the pieces are drawn from one shared library, so a card that turns out to be piece A is a card that was not piece B. The pieces compete for the same draw slots, which makes them negatively correlated, and the true joint probability comes in below the product rather than above it.

The gap is small but it runs the wrong way for a deck builder. Two pieces at four copies each in a 99-card deck, wanting two of one and one of the other by turn five: the product says 9.32% and the real answer is 7.96%. The shortcut flatters the deck every time.

The combo tool shows both figures side by side for that reason, along with each piece on its own, so the distance between "each piece is likely" and "all of them together are not" is visible rather than asserted.

The pieces must share no cards. A card that fills two roles has to be counted in one of them. The calculator takes counts rather than card names, so it cannot detect an overlap, and a double-counted card produces a figure that is too high. This is the one input error the tool cannot catch for you.

Format rules

Two rules change the answer, and the format control sets both.

Drawing on your first turn decides how many cards turn one is. In multiplayer Commander nobody skips their first draw, so turn one is eight cards and turn T is 7 + T. A 60-card player on the play skips it, so turn one is seven cards and turn T is 7 + (T - 1).

The free first mulligan decides how many cards a mulligan costs. In Commander the first one costs nothing, so one mulligan still keeps seven. Everywhere else the first mulligan already puts a card on the bottom.

The library size is what you draw from, so a Commander deck is 99: the commander starts in the command zone. Both figures are defaults you can change.

What the mulligan model does

The draw odds and combo tools model mulligans. It assumes you mulligan every hand that misses your question and keep every hand that hits, up to the number of mulligans you allow. The hand you play is therefore the first one that hit, or the last one you were dealt if none did, and the figure sums over which of those happened.

Two details matter for reading the number.

Cards you bottom leave the sample. Under the London mulligan you draw a fresh seven and put cards on the bottom to pay for it. They sit under everything you draw for the rest of the game, so the model treats them as gone.

You bottom the cards you did not want. Asking for two lands, a seven-card hand holding four of them can bottom two non-lands and lose nothing. This is why allowing a mulligan often costs less than people expect, and why the free first mulligan changes the figure only when the hand you keep cannot hold everything you asked for.

Allowing more mulligans always raises the figure this model reports, because nothing here charges you for the card disadvantage, the lost tempo, or the hands you would actually keep for reasons the question does not describe. Reading five mulligans as better than one is reading the model past what it knows.

What none of them count

Cards that draw cards. A cantrip, a wheel, and an impulse effect all change how many cards you have seen by turn T, and the tools count only your natural draw for the turn. A deck with real card draw beats these figures.

Tutors. A tutor turns one card into whichever card you need. Counting it as one more copy of the group is closer to right than ignoring it.

Whether you can cast what you drew. Every figure counts cards in hand. Mana, timing, and whether an opponent lets the spell resolve are all outside it. This matters most on the combo tool, where holding the pieces and assembling them are different achievements.

Whether a land enters untapped, or a card is castable. The land drop tool counts lands. A tapped land, a land that enters untapped only under a condition, and an untapped dual are one land each to it.

Conditions multiplying. Every figure answers one question. A plan that needs three separate 80% pieces comes in at 51%, and nothing on these pages will tell you that. Multiply them yourself.

Making every land drop in a row. The land drop tool reads each turn on its own. The chance of making all of them is lower than the lowest figure in the column, because the turns are not independent.

Mean and median mulligans

The keepable hand tool reports two figures for how deep you go, and they are different statistics with different meanings. Reporting one under the other's name, or averaging them, is a mistake worth naming because published calculators in this space do it.

The mean is the long-run average, (1 - p) / p where p is the chance a fresh seven keeps. The median is what happens half the time, ceil(log 0.5 / log(1 - p)) - 1. The distribution has a long tail, so the mean sits above the median, and the gap widens as the keep rate falls:

Both are exact rather than estimated, and the reason is the London mulligan: every new hand is a fresh seven off the full deck, and you decide whether to keep before anything goes to the bottom. So the keep chance is the same on every re-roll, which is what makes the number of mulligans a geometric distribution with a closed form.

No mulligan penalty

Some calculators apply a flat penalty, commonly 20%, to hands kept after a mulligan, standing in for the cost of playing with fewer cards. Nothing here does that. The weighting has no derivation behind it, and a number with no derivation is worse than no number.

The cost of going deeper is shown as the card count at each step instead. A condition asking for three specific cards cannot be met by a hand bottomed to two, and the tool marks those rows rather than discounting them, because that is a fact about the hand rather than a guess about its value.

The probability bands

The draw odds tool shades four bands behind the chart. They are rules of thumb from common practice, not results:

Use your own judgement for your deck. A combo deck and a grindy deck disagree about what an acceptable figure is, and both are right.

Getting this against your own deck

Build a deck in the app and open its analysis tab. The odds panel there is this same calculator with your list already loaded: your categories become the groups, the library size is your card count, and picking a group fills in its copy count. That is the version worth using once you have a list, because the question you want to ask is usually about a category you have already built.