I hope this will be a bit helpful for some of you out there who, like me, lack the math skills to compute the probability of drawing the crucial 3 towers in my opening hand. The answer is Hypergeometric Probability. Don't run away yet, there's a handy calculator here for just this thing:
http://stattrek.com/Tables/Hypergeometric.aspx
The calculator itself is simpler to use than you might think.
Population Size: The number of cards in your deck. Simple enough.
Sample Size: The amount of cards you are drawing. Usually 7 for the beginning of the game, but you can increase that to figure out how many extra turns as well it will take.
Number of Successes in Population: Total number of cards you want to draw.
Number of Successes in Sample: Amount of the card you want to draw.
It will then give four outputs.
Hypergeometric Probability (P X=1): Simply put, the probability that you will draw your Number of Successes in the Sample Size, exactly, no ifs, ands, or buts.
Cumulative Probability (P X<1): The chance that you will draw less than 1 (1 in this case, its technically the Number of Successes in Sample) of your wanted card.
Cumulative Probability (P X<=1): The chance that you will draw less than or equal to 1 (1 in this case, its technically the Number of Successes in Sample) of your wanted card.
Cumulative Probability (P X>1): The chance that you will draw greater than 1 (1 in this case, its technically the Number of Successes in Sample) of your wanted card. Note that the probability of drawing 1 card is NOT included in this number. For that, see below.
Cumulative Probability (P X>=1): The chance that you will draw greater than or equal to 1 (1 in this case, its technically the Number of Successes in Sample) of your wanted card.
As one example, I'm going to dig up the scenario where I want to see what the probability of drawing at least 1 Sundial in my opening hand in my old Dive deck. Therefore, I fill it out like so.
Population: 30
Sample: 7
Number of Successes in Population: 3
Number of Successes in Sample: 1
The answer was slightly surprising.
Hypergeometric Probability (P X=1): ~.436, or roughly 44%. 44% chance of drawing a Sundial in my opening hand.
Cumulative Probability (P X<1): Also ~.436/44%. Basically, an equal chance of drawing 1 or 0 Sundials
Cumulative Probability (P X<=1): ~.872/87%. This number is rather useless as this scenario does not guarantee a Sundial despite the high probability shown.
Cumulative Probability (P X>1): ~.128/13%. I have a 13% of drawing more than 1 Sundial in my first hand.
Cumulative Probability (P X>=1): ~.564/56%. This is the stat I'm specifically asking about. I now know I have a 56% chance of drawing at least one Sundial in my opening hand. Not bad, but not good.
This is another example but with Quantum Towers and my Sloth Poison deck; its not particularly necessary and is just another example with the numbers changed.
Population: 60. Its a big deck.
Sample: 7. Still want to know about my opening hand.
Number of Successes in Population: 20. 1/3 of my deck are Towers, but that doesn't guarantee a 1 in 3 chance of drawing those Towers.
Number of Successes in Sample: 3. I want to know the probabilities involved in drawing 3 or so Towers.
Hypergeometric Probability (P X=3): ~.270/27%
Cumulative Probability (P X<3): ~.570/57%
Cumulative Probability (P X<=3): ~.841/84%
Cumulative Probability (P X>3): ~.159/16%
Cumulative Probability (P X>=3): ~.429/43%
End result, I have a 42% chance of drawing at least 3 Towers or more in my opening hand, but a 58% chance of drawing less. Odds are not great, but good to know. Hopefully this little post can be used, especially with QI to help design some better decks.
Also, if anyone feels my math is off or something like that, please call me out. I'm reasonably sure about these numbers, but I was never great at statistics.
Thanks to Daxx for giving me the link months ago.
Part 2: Graphs!
I finally got off my rear and coded up some 4D graphs to demonstrate the trends in the probability. I don't think they reveal too much unexpectedly, but they are cool to look at. Included in the program is a way to implement and properly use the mulligan rule, and plotting every single valid probability. This is cumulative probability here, so one blue dot in roughly the spot of drawing 2 cards in a 30 card deck having say 5 of that card means that there is a blue probability (~10%?) of getting at least the required 2 cards. Also, the color bar shows probability via percent; that 60 is a 60% chance of that event happening.
The first five figures are with mulligan on, using 12 pillars and showing stats for an opening hand. The sixth image has mulligan disabled, pillar count unused, and the same opening hand. Graph seven just calculates with mulligan, 12 pillars, out to the third card drawn, or ten total.
Disregard, improperly rendered graph
