B · Engine & Sequencing · §8
A fully assembled natural combo appears in one opening hand in seven. Everything about how this deck is built follows from that number.
The relevant maths is hypergeometric: the chance of at least one copy in a sample, where N is the deck size, K the number of relevant cards and n the cards seen.
P(at least one) = 1 − C(N−K, n) / C(N, n)
At least two lands, 78.3%. Two to four lands, 73.8%. At least one , 39.9%. At least one dedicated target, 54.1%. At least one of the eleven direct enablers, 77.8%. Goryo plus a dedicated target, 20.0%. Goryo plus target plus a direct enabler, 13.8%. , 39.9%. Force plus another blue card, 19.6%. plus another white card, 32.0%.
These are raw pre-mulligan numbers and they do not count surveil lands as guaranteed enablers.
The construction already assumes this. Frog finds or discards pieces, Archaeologist and Gaze see extra cards, surveil lands improve access, and removal buy time, and the fair threats convert incomplete combo hands into games you can win.
Under a simplified model where one has already left the deck and the remaining fifty-nine cards are available, the chance its top ten contains at least one card of each main-deck type is roughly: land 99.2%, creature 98.2%, instant 96.1%, sorcery 61.9%. The expected number of distinct types is about 3.55.
That distributes as four types 57.0% of the time, three types 41.4%, and two types 1.6%. Real games vary because cards have already moved to hand, battlefield, graveyard or exile.