At first I started out doing all that 60 choose 7 stuff, trying to act smart with all the college probability stuff I pretended to pay attention to, but really, Like dmaster said, it depends on the number of basics, because the more basics in your decks, the less chance you have of getting a Sableye or a mulligan.
But theoretically, I think this works:
****If you're not very mathy, or haven't studied Discrete Mathematics or Probability and Statistics yet, jump to the end of this post.*****
I will use C(n,r) to denote "n choose r," which of course equals n!/r!(n-r)!
The best way to approach this is to abuse the concept of complements of probability. First of all, let's take a look at the probability of not getting a Sableye at all, regardless of the other cards in your deck:
C(4,0)C(56,7)
------------- = 0.6005
C(60,7)
So, you have a 60% chance of not getting a Sableye in a seven-card hand, which conversely means that you have a 40% chance of starting with at least one Sableye in your hand. This also says that if your 4 Sableye are the only basic Pokemon in your deck, you have a 60% of getting a mulligan XD
Now what this doesn't factor in is the chance you have of getting another basic Pokemon instead of Sableye. To figure that, let's let b denote the total number of basic Pokemon in your deck (including Sableye). Then to determine the chance of getting a mulligan, you would do the following:
C(b,0)C(60-b,7)
----------------
C(60,7)
As an example, let's say your deck has 12 basic Pokemon. Then by the above formula, you have about a 19% chance of getting a mulligan, which conversely means you have an 81% chance of drawing a basic in your starting hand. Pretty good, right?
Now coming back to Sableye, let's say that 4 of those 12 basics are Sableye. Now, there's probably an easier way to do this, but this is how it makes the most sense in my head. Out of that 81% chance of drawing a basic, take out the combinations in which there are no Sableye.
P(no Sableye, but have basics) =
[Summation from x = 1 to 7 of C(4,0)C(12-4,x)C(48,7-x)]
------------------------------------------------------- = 0.409854
C(60,7)
This means that out of the 81% chance that you will start with a basic, there is a 41% chance that a Sableye will not be in that hand. Therefore, in conclusion, there is a 40% chance that you will have a Sableye start with 4 Sableye and 8 other basics, which is the same probability we came up with in our first calculation.
*****************************************************************************************************
So for those who skipped over the above math stuff, to answer his question in the most simplest of terms, if you have 4 of any card in your deck, you have a 40% chance of drawing it in your initial 7 cards. But this is very misleading.
If your only basic Pokemon are your 4 Sableye, then you have
- a 60% chance of getting a mulligan.
- a 100% chance of getting at least one Sableye if you don't mulligan.
Whereas if you had 4 Sableye and 8 other basic Pokemon, you would have
- only a 19% chance of getting a mulligan.
- a 59% chance of getting at least one Sableye if you don't mulligan.
See the difference? Obviously, the more basic Pokemon you have in your deck, the less chance you have of getting a true "Sableye start."
Interestingly enough, if of those 12 basic Pokemon you had 1 Unown G, you would have
- about a 12% chance of having Unown G in your starting hand.
- about a 3% chance of it being the only basic Pokemon in your hand!
- LOL, srsly
If anyone is interested, I can post formulas for figuring these stats out.
Heh, that post took so long to come up with, I didn't see the two posts above me
But yeah, same thing as above.
