I give a sincere nice try on the math Ryan. I have degree in Actuarial Science and earn my living as credentialed actuary. To get credentials, I had to pass professional exams, the 2nd of the 10 exams had included nasty nasty probability problems like this on them. Let's just say, I didn't get the problems correct the first time I had to do them. The fact that you are close is impressive, but I feel the need to correct this.
The odds of a mulligan with a 16 basics ins a 60 card deck is correct at 9.9% (10%) Your math is 100% on that.:thumb: The 90% of the universe of 7 random cards are non-Mulligans. We need to remember this because the 10% of the mulligan universe we don't want to include.
:fire:There are multitude of problems that you have in calculating the lone 30HP basic start: The math ignores the probability of the multiple Hopips. The 47.45% is the probability of having 1 or MORE 30HP pokes, in your seven card draw. The multiplication of the two probabilities don't work because the total lack of independence and order dependence issues.... to say it politely, the approach was wrong.
To get your mind into the correct math construction, let's just expand on your probabilities of having a mulligan. There is only one unique scenario, card one, card two, .. Card seven are all non Basics. One scenario with the probability string as you stated as:
(44/60)*(43/59)*(42/58)*.....(38/54) = 9.9%.
This is the probability of having drawing 0 basic in 7 cards in a 60 card deck with 16 basics in the deck with replacement. Hypergeometric Distribution in Excel. In Excel =HYPEGEOMDIST(0,7,16,60). You did this right.
What is the probability of having only 1 basic of the 16 basics in a 7 card draw of a 60 card deck with replacement. It is really made up of 7 scenarios that gives you the one basic start. The scenarios are defined by what card # you get the basic, and the others are all non basics. Thus, the First scenario is drawing a basic on the first card, then next 6 cards being non basics. The Second Scenario is draw non basic card one, then a basic card two, and rest are non basics.. Third Scenario, the 3rd card is the basic,... so on and so on.
Probability of Scenario 1 is (16/60)*(44/59)*(43/58)*(42/57)*(41/56)*(40/55)*(39/54) = 4.18%
Probability of Scenario 2 is (44/60)*(16/59)*(43/58)*(42/57)*(41/56)*(40/55)*(39/54) = 4.18%
Probability of Scenario 3 is (44/60)*(43/59)*(16/58)*(42/57)*(41/56)*(40/55)*(39/54) = 4.18%
..
Probability of Scenario 7 is (44/60)*(43/59)*(42/58)*(41/57)*(40/56)*(39/55)*(16/54) = 4.18%
Thus the probability of drawing 1 basic in a hand of 7 is 29.24% = (7 * 4.18% ).
In Excel, = HYPERGEOMDIST(1,7,16,60).
Since Mulligans are redo's. The probability of starting iwth 1 basic, given you have 1 or more basics need to be calculated.
1 Basic Start = Prob (Basics = 1 ) / Prob (Basics >= 1)
1 Basic Start = Prob (Basics = 1 ) / [1 - Prob (Basics = 0)]
Thus Prob (Basics = 1) = 29.24% and Prob(Basics >=1) = 100% - 9.92% = 90.08%
1 Basic Start = 29.24% / 90.08% = 32.47%
Well given that we have started with only 1 basic 32.47% of time and we know that 5 of the 16 basics are 30 HP, it is a straight forward multiplication of 5/16.
Thus lone 30HP pokemon start is 10.15%, not 12.7%.
You could skip the 5/16 part of the math and add up the 7 scenarios of draw 1 of the 5 bad basics and 6 non basics(44 cards). But it still needs to be divided by the universe of non-mulligan starts. (If you look at the algebra above, you are just replacing the 16 with a 5 directly in the string of multiipcations)
tl;dr lol. I agree with your main point of your post. Is there something wrong with 4 call and 3-4 Claydol for consitency btw?