The term you're looking for is
hypergeometric distribution.
To solve these, you need to know the standard
combination without repetition formula:
[sub]n[/sub]C[sub]r[/sub] = n! / (r! (n - r)!)
Where n is the total size of your set, and r is the number of selections you're making.
Now let's define the following variables:
x = number of a particular card (or type of card) you have in the deck
d = number of cards in the deck (always 60, in Pokemon's case)
z = number of cards you are drawing (7, in the case of your initial hand)
s = number of successes (how many of that particular card you're wanting to get)
And since we can define these questions as a binomial probability (either we got the card we wanted or we didn't), we can simplify the formula for this hypergeometric distribution to:
[sub]x[/sub]C[sub]s[/sub] * [sub](d - x)[/sub]C[sub](z - s)[/sub] / [sub]d[/sub]C[sub]z[/sub]
But that gives you the probability of drawing
exactly s. If you're wanting to know the probability of getting
at least 1 of something, it's easier to let s = 0 (calculate the probability of
not getting the desired cards) and then subtract that probability from 1.
So let's answer the three questions posed already:
What is the probability of drawing a basic in my hand of 7 if I have 12 basics in my deck?
[sub]12[/sub]C[sub]0[/sub] * [sub]48[/sub]C[sub]7[/sub] / [sub]60[/sub]C[sub]7[/sub]
= 1 * 73629072 / 386206920
= 0.1906466927
1 - 0.1906466927 = 0.8093533072
So an 81% chance of drawing at least 1 basic in your initial hand with 12 in the deck.
Out of that what is the possibility of only drawing 1 basic in my hand of 7?
[sub]12[/sub]C[sub]1[/sub] * [sub]48[/sub]C[sub]6[/sub] / [sub]60[/sub]C[sub]7[/sub]
= 12 * 12271512 / 386206920
= 0.3812933854
38% chance of a one-basic start
Out of that what is the probability of that one basic being an absol if i have 3 absol in my deck?
Well, if 3 out of 12 Basics in your deck are Absol, that's 25%. So 25% of your one-basic starts will be Absol.
0.3812933854 * 0.25 = 0.0953233464
Roughly 9.5% overall chance of starting with Absol only.