How likely you are to have the cards you need in your opening hand, or by a given turn, in a 99-card Commander deck.
How this is worked out
It’s the hypergeometric distribution: the exact chance of drawing a given number of particular cards from a shuffled deck, with no guessing or simulation. Looking for several kinds at once, like two lands and a piece of fast mana, works the same way across all of them together.
Your opening hand is 7 cards and each turn you draw one more. In a pod of three or more players nobody skips their first draw, so by your turn 3 you’ve seen 10 cards. In a duel, the player going first skips it. Your commander starts in the command zone, so it isn’t in your library: that’s 99 cards, or 98 with partners or a background.
Each mulligan is a fresh shuffle and a new seven, so the chance across hands is worked out hand by hand. The first mulligan is free in multiplayer. After that you put one card on the bottom for each mulligan, so a hand only counts if it can still keep what you need.
When you mulligan looking for something, a hand without it goes back, and the hand you finally keep is played out to the turn you asked about.