How the hypergeometric calculator works
What the numbers of the calculator mean, when they are exact and when they come from simulated games.
The hypergeometric distribution
Drawing from a shuffled deck is drawing without putting cards back: every card you see changes what is left. The hypergeometric distribution counts exactly how many ways a group of cards can end up among the ones you see. With 4 copies of a card in 60 cards and 7 cards in the opening hand, the chance of at least one copy is 1 minus the ways to draw 7 cards from the other 56, divided by all the ways to draw 7 from 60: about 39.9%.
The calculator does this for up to four groups at the same time, also when a card counts for more than one group (a dual land is a source of two colors), with a minimum, a maximum or both.
Turns, play and draw
By turn 3 you have seen your opening hand and one card per turn, minus the draw that the player who starts skips in a two-player game: 9 cards on the play, 10 on the draw. In multiplayer games nobody skips that draw. Extra cards drawn (cantrips) add to the count; cards looked at with scry or similar are counted as seen, which overestimates a little.
Mulligans
With the London mulligan you draw 7 new cards and put one on the bottom for each mulligan. The calculator tries every possible hand of 7 and keeps the cards you need, putting first on the bottom the copies beyond a maximum, then the cards of no group. It is a fixed rule, close to how people play, not the best choice in every case.
Exact odds and simulation
Exact odds only work when the number of cards you see is fixed. Cards like Ponder, Expedition Map or Ancient Stirrings change what you see depending on your mana and your choices, so the Simulation tab plays thousands of games turn by turn with fixed rules instead, reading what each card does from its Oracle text. Without those cards the simulation lands on the exact odds; with them it shows how much they help.
The margin of error
A simulation of 20,000 games gives a result like 48.9% ± 0.7 points: the true value is in that range 95 times out of 100. Two results closer than their margins can be the same thing, so compare changes that move the odds by more than that.