There's an old and very bad thread on the board which I don't want to Necromancer. In it, eHalcyon makes some claims about probabilities. Here I do the math to see if those claims are accurate.
If you had five 5 coppers and 5 estates, there is just as much chance of you drawing ccccc as there is of you drawing ccece, even if one "looks" more random to the human mind.
If we put little stickers on the cards labeled "1" through "10", there would be 10! different orderings. But since we don't distinguish between the 5 estates nor the 5 coppers, there are only 10! / (5!*5!) = 252 = 2*2*3*3*7 orderings.
How many of those orderings start with "ccccc"? As many as there are (non-distinguished) ways of ordering the remaining cards, which is just 1 (that ordering being "eeeee").
The number of orderings that start with "ccece" equals the number of orderings which end with some reordering of "cceee". There are [5 choose 2] = 10 such orderings.
Hence, you're 10 times as likely to draw "ccece" than you are to draw "ccccc". You are even more likely to draw "3 coppers and 2 estates in any order" as you are to draw "ccece" in that specific order, because there are more orders (and all orderings are equally likely).
[D]rawing a hand of all actions, then a hand of all treasures followed by a hand of all greens seems really non-random to us. But with perfect randomness, there is just as much chance of drawing these clumps as there is of getting three hands each with mixes of all three.
Restated, if you draw three batches each of five balls from a jar with five green, five white and five yellow balls, not putting the balls back between draws, you're equally likely to draw three monochromatic batches as you are to draw three rainbow batches (all colors represented).
The total number of orderings is 15! / (5!*5!*5!) = 756756.
There are six orderings of monochromatic batches: GWY, GYW, WGY, WYG, YGW, YWG. (This is shorthand for ggggg+wwwww+yyyyy etc.); 6 out of 756756 is a bit less than 1 in 100 000.
A rainbow batch has gwyXZ in some order, where X and Z are both members of {g, w, y}. X and Z can either be equal or different. This means that a rainbow batch either has three of one color and one of each of the rest, or two pairs and one loner.
Suppose the first batch is rainbow, with a color occurring thrice; let's say it's green, so it's gggwy in some order. There are 2g+4w+4y left. If all batches are rainbow, the remaining two batches are either [gwyyy+gwwwy], [gwwyy+gwwyy] or [gwwwy+gwyyy].
Note that gggwy has 5! / (3!*1!*1!) = 20 orderings, and gwwyy has 5! / (1! * 2! * 2!) = 30 orderings.
So there are 20 * (20*20 + 30*30 + 20*20) = 34000 orderings that start with gggwy. But picking green was arbitrary; there are just as many orderings that start with wwwgy or yyygw. (Having picked green, we shouldn't count the swap of white and yellow as separate, because that would double-count the gwyyy+gwwwy and gwwwy+gwyyy follow-ups. It would also double-count gwwyy+gwwyy, because it isn't different from gyyww+gyyww.)
So that's 3*34000 = 102000 orderings that start with a rainbow batch with one color repeated thrice.
If instead it starts with two pairs, let's say gwwyy in some order, the rainbow follow-ups are [gwwyy+gggwy], [ggwwy+ggwyy], [ggwyy+ggwwy] and [gggwy+gwwyy], where all hands can be reordered.
That's 30 * (30*20 + 30*30 + 30*30 + 20*30) = 90000 orderings. But this again is with green arbitrarily picked to be the starting loner, counting those with a white or yellow loner first (and the other two colors paired) gives us 3*90000 = 270000 orderings.
In total, that's 102 000 + 270 000 = 372 000 orderings out of 756 756, or approximately half.
So rainbow batches are about 1 in 2 compared to monochromatic batches which are about 1 in 100 000, so rainbow batches are about 50 000 times as common.
ConclusionI think eHalcyon's quoted claims are incorrect. Here are some similar claims which I think are correct:
- The ordering ccccc eeeee is just a likely as the ordering ccece eceec.
- The ordering ggggg wwwww yyyyy is just as likely as the ordering wggyg wgwyw ywyyg.
I agree that monochromatic hands feel 'less random', whatever the hell that means, to most people (including me). I think eHalcyon's point is accurate, but slightly misstated, which in turns means that this whole thread is a bunch of pedantic nitpicking.
Am I doing f.ds'ing right?
