Question about statistics

[tinkthank]

Just playing my first game of Arco (thought I would try out some races I have never used before), and it took me 6 turns to get a Mystic without all three random paths stacking. (That is, I recruited a w3, w3, e3, w3, f3, e2a1 Mystic in that order.) Until I got the e2a1 mystic, I was about ready to think that the elementals ALWAYS stack.
I am obviously both stupid and have bizarre luck. (It took me 10 months to get a Starspawn with 2 stacking randoms and I still dont understand the formula.)
Can someone please list here the formula for figuring out the chances of things like this? I had stats back in high school but since then I have left the field of natural sciences and moved into the armchair humanities, and I just dont get it.
Is it something like 7/8*7/8*7/8 for the mystics? Damn, sorry, and thanks in advance.

[Chazar]

Hmm, I dont know the mystic by heart, but if they have 3 random elemental picks, it goes like is:

1. There are 4 elemental paths, so for each pick there are 4 possible results. The chance that I get a desired is 1/4. (Also see notes at 5. below.)

2. We want to know the probability for 3 stacked picks, but in any elemental paths, so the first pick will always be ok, whereas the other two must match the first:
1 * 1/4 * 1/4 = 1/16 = 6,25%. Only one out of 16 mystics will have all three picks in any one elemental path.
(Another more detailed way of reasoning could be: the probability of a three-stack in a certain path is 1/4 * 1/4 * 1/4, but since there are 4 types of mystics with such a three-stack, we can multiply this probability again by four, yielding the same result as before.) I'd say you're lucky to get a three-stacked mystic!

3. The probability to not have a single stacked random is like this: The first is always fine, since it never stacks. For the second pick not to stack, it must be one of the three remaining elemental paths, and for the Last, only two good choices remain:
1 * 3/4 * 2/4 = 3/8 = 0,375%, i.e. more than a third of all mystics are pretty versatile, and there are four types of such versatile mystics.

4. Either move somewhere else or try the lottery: The local randomness grid of your area seems to be malfunctioning...

Edit:
5. Disclaimer:
What I've said in 1., that the chance to obtain any desired pick is 1/4, is truly an assumption of mine. I do not know if this is true, since Illwinter could be evil and have coded the picks with unequal probability chances in some weird circumstances intentionally to confuse some players. Furthermore, we must consider that true random generators are too expensive, so do not expect your average computer to be capable of true random number generation. All it can provide is pseudo-random number sequences, but as long as you are not producing TANs for Online banking, we should not worry to much about it. The pseudo-random numbers are random enough for household use.

All in all, we can expect to observe the probability of any given specific pick to be 1/4 (i.e. Illwinter is nice) and that this is practically independent of any previous picks choosen or observed (i.e. true randomness).

[Zooko]

The thing about probability is that it only applies to unknowns, e.g. to the future.

The chance that you got five Mystics in a row with 3-stacks is 1 out of 1. It's what happened!

That might sound flippant, but it is a common mistake to attempt to apply probability reasoning to selected events from the recent past. Note the "selected" part.

Also, in a truly random sequence, patterns will appear. What's the chance that people summoning a string of mystics will see something unusual about the pattern, like never getting any fire magic, or every mystic having at least one air magic, or something. The odds are pretty high!

[Chazar]

@Zooko: Well, I thought he had asked about the probability to encounter that particular pattern again, which is of course independent of past experiments...

@thinktank: BTW, the probability to observe exactly the same sequence of mystics (or pattern) is
((1/4*1/4*1/4) *1)^5 * ((1/4 * 1/4 * 1/4) *3)^1 = something small
The 3 is there because there are three types of a1e2 mystics (a either in the first, second or third pick obtained).

[tinkthank]

Hey thanks a lot!

[ckfnpku]

Don't atlantian kings also tend to stack their randoms? I think these units have some mechanism that stacks their randoms.

[johan osterman]

ckfnpku said:
Don't atlantian kings also tend to stack their randoms? I think these units have some mechanism that stacks their randoms.

Their random allways stack. They have 2 random instead of 1 random + 1 random, so to speak.

[Chazar]

Aha, as I had suspected: There is at least one special case!