Magic The Gathering: probability of casting a 3 drop on turn 3
There is a legit mathematical answer to this question, but it's a bit outside my ability. What I can offer you, however, is an example of how Magic players of various levels of mathematical knowledge have attempted to tackle the question.
This article by Frank Karsten is a good start. Doing it mathematically, you get stuff like this:



Eventually, Karsten just gives up and decides to simulate it. A sample of his conclusions for 60-card decks:

To have a good (90%) chance of casting a one-drop (e.g. Duress) on turn 1: 14 sources of its color.
To have a good chance of casting a three-drop with one colored mana symbol (e.g. Thirst for Knowledge) on turn 3: 12 sources.
To have a good chance of casting a three-drop with two of the same colored mana symbol (e.g. Liliana of the Veil) on turn 3: 19 sources.

Unfortunately, this particular analysis doesn't focus too much on multi-color non-land cards (more on multi-color lands, which are invaluable for building competitively powerful decks).
He also recommends looking at preexisting competitive decks as a guide, since players tend to build for consistency.
A 2012-season Delver deck is probably a good guidepost, since it's a deck that played Geist on-curve and as one of its most color-demanding cards.
How many white sources is it packing?
lands – 22:

9 Island
2 Plains
4 Glacial Fortress
3 Moorland Haunt
4 Seachrome Coast

card-filtering — 9 cards (plus the Snapcasters):

2 Thought Scour
3 Gitaxian Probe
4 Ponder

So that's 10 white mana sources, plus 4 card-filtering cards that dig deep (the Ponders). Pretty close to the 12 that Karsten said you want to get a 90% chance of casting a single-color three-drop on turn 3.