What's the probability of having a combo on the first turn in MTG?
You can do this calculation using the multivariate hypergeometric distribution. The setup is as follows:
The deck of 60 cards consists of: 4 cards of type A, 4 cards of type B, 4 cards of type C, 4 cards of type D, and 44 cards of type E (other).

Your criteria are that a hand of 7 cards contains at least 1 card of type A, at least 1 card of type B, at least 1 card of type C, and at least 1 card of type D.

For a given hand arrangement, you can calculate the probability using the formulas in the link. As an example, the probability of the hand (1 card of type A, 1 card of type B, 1 card of type C, 2 cards of type D, and 2 cards of type E) is:

(4 choose 1) * (4 choose 1) * (4 choose 1) * (4 choose 2) * (44 choose 2) / (60 choose 7) = ~0.000941.

Note that this probability is for this specific hand, and there are many that meet your requirements. You will want to make a table of all realizable hands and sum up the probabilities. (or alternatively a table of all hands that don't meet the criterion, and subtract the sum from 1).

=== table of realizable hands ===


1,1,1,1 of (A, B, C, D), 3 of other


(4 choose 1) * (4 choose 1) * (4 choose 1) * (4 choose 1) * (44 choose 3) / (60 choose 7) = ~0.00879


1,1,1,2 of (A, B, C, D), 2 of other [4 variants]


4 * [(4 choose 1) * (4 choose 1) * (4 choose 1) * (4 choose 2) * (44 choose 2) / (60 choose 7)] = ~0.00376


1,1,2,2 of (A, B, C, D), 1 of other [6 variants]


6 * [(4 choose 1) * (4 choose 1) * (4 choose 2) * (4 choose 2) * (44 choose 1) / (60 choose 7)] = ~0.000394


1,2,2,2 of (A, B, C, D), 0 of other [4 variants]


4 * [(4 choose 1) * (4 choose 2) * (4 choose 2) * (4 choose 2) / (60 choose 7)] = ~0.00000895


1,1,1,3 of (A, B, C, D), 1 of other [4 variants]


4 * [(4 choose 1) * (4 choose 1) * (4 choose 1) * (4 choose 3) * (44 choose 1) / (60 choose 7)] = ~0.000117


1,1,2,3 of (A, B, C, D), 0 of other [12 variants]
12 * [(4 choose 1) * (4 choose 1) * (4 choose 2) * (4 choose 3) / (60 choose 7)] = ~0.0000119


Total Sum of above = ~0.01307