Kayle: I will outline some basic math computations and "plug and chug" formulas.
I have most of the numbers memorized (1 card=12%, 2=22%, 3=31, 4=40), but sar86 points out how to do this for numbers greater than 4, which is important when you start doing advanced probability and want to calculate TOTAL outs for say- energy (roseanne+cyrus+energy) or a stage 2 out (pokemon+luxury ball+bebe+communication(+pokemon)), etc etc.
To figure out the odds of something happening, we can find out how often it won't happen (hence the 1- X*Y*Z...)
So, probability of energy= 1-odds of NOT having an energy
NOT having an energy is determined for each of the 7 cards we have.
If we start with 6 energy, then on the first card we draw, there is a 54/60 chance of it not being an energy, but as each card is drawn, this probability slightly boosts. We multiply them all together to get the collective chance of the entire 7 card hand NOT having an energy.
So, if we start with 54/60, when we draw another card- if we assume no energy, it is 53/59, and so forth, until you get:
1 - (54/60)*(53/59)*(52/58)*(51/57)*(50/56)*(49/55)*(48/54)
So, if the probability of NO energy is 45.9, then the probability of YES energy is 1-45.9 or about 54%. That number is why I instantly knew that 8/10 for an energy in opening hand in jumpluff is absurd.
To get that kind of number, like I said, you'd be running 11 or 12 energy- twice as many energy! This also shows the non-geometric progression of the distribution. Although the energy between 6 and 12 is doubled, doubling the energy only boosts the consistency of opening with an energy from 54% to 81%, a 27% increase. The first 6 energy gave a 54% increase, and the second 6 gave HALF of that.
This is what I was trying to get at in the article itself when I suggested that as consistency is added, its overall INFLUENCE on the probability DECREASES. That is why there is an ideal ratio/threshold to hit in order to get an ideal amount of consistency for how much deck space it is taking up- which hurts other parts of the deck.
I will later outline my method of finding the odds of a SPECIFIC basic starter (useful to know how often you will get that Regice start, or Spiritomb start, etc).
1-(N-X/N)*([N-1]-X/[N-1])* and so forth, until you get N-6 for a 7 card opening hand, or N-7 for an 8 card opening hand (after you draw for turn)
N denoted the number of cards in the deck left, and X denotes the number of "outs" we are calculating with.
I will outline odds of a supporter start, odds of a specific starter, odds of a turn 2 claydol given certain other combinations- odds of a turn 2 bright look, etc. Calculating a specific starter uses the same kind of probability calculations, but also takes into consideration mulligans and starting with multiple basics and so forth.
I remember last year when I made my LuxTrix deck, I figured out the exact odds of a turn 2 bright look. Calculated the odds of luxray start, non-luxray start, odds of drawing a rose/Q with my non-luxray start (or having a free retreat electrike or crobat), the odds of getting the energy gain+ lightning, or cyrus, the odds of a bebe/etc. It took a while to do, but it was definitely enlightening to see how my deck ACTUALLY should function and knowing just how many resources it takes to reasonably expect certain scenarios to take place.