Party Quests APQ QoL: Bonus Room Spawn

I spawn on top of other people 80% of the time.

I think spawning at a fixed spot (center, or top center) would be the best QoL. It would also be the easiest option to implement.

This would benefit everyone, people with mobility can call the more out-of-reach spots, letting the guys without tele/FJ choose the spots in the center or bottom.

 

Under the assumption that each member gets a unique spawn point.

In the following problems, we assume that 6 party members participate in the bonus stage, and that the spawn points are assigned in a truly random fashion (each with equal probability).

There are two ways of looking at this problem. You can refer to this source. Let P(share) denote the probability that at least two party members share a spawn point.
View 1: Party members are distinguishable and spawn points are distinguishable. P(share) ≈ 0.985

Spoiler: Explanation

In this case, there are 6 members, each of which have 6 choices for a spawn point. That's

6*6*6*6*6*6 = 6^6 = 46,656 unique arrangements.​

A quantity one might be interested in: What is the probability that at least two people will share a spawn point? Well, without going into too much detail, the easiest way to calculate this is to compute 1 minus the probability that everyone has a unique spawn point. The probability

P(Unique) = (# unique arrangements)/(total # arrangements).​

We know the denominator, 46,656. The numerator is obtained by observing that the first member has 6 choices, then the second has 5, and so on. This gives 6*5*4*3*2*1 = 6! = 720, the number obtained in the above quote. In total we have

1 - P(unique) = 1 - 720/46,656 ≈ 0.985, ​

the probability that at least two members will share a spawn point. Very high indeed.

View 2: Party members are indistinguishable and spawn points are distinguishable. P(share) ≈ 0.998

Spoiler: Explanation

Here we use the balls and urns, where the players are balls and the urns are spawn points. We have

nchoosek(6 + 6-1, 6-1) = nchoosek(11,5) = 462 total arrangements.​

In this scenario, the players are indistinguishable, that is, imagine we are distributing six white balls into six urns, each urn a different color. There is only one way to distribute the balls where each urn gets exactly one ball. With our "1 minus" trick from before, we have 1 - 1/462, which is practically probability 1.

Why did we care whether party members are distinguishable?
You can imagine a party in which every participant is a high mobility class (either a mage or NL), which would be much closer to indistinguishable, as opposed to a party with one mage, one NL, a corsair, and three warriors.

A Quick Understanding
If you want a quick and dirty way to understand this problem, you'll notice that we are exactly one spawn point away from invoking the pigeonhole principle. That is, if we have six party members and five spawn points, then no matter what, there will be a spawn point with at least two people on it. Having six spawn points doesn't necessarily provide much wiggle room.

Thanks for reading.
Of course, take this all with a grain of salt for several reasons. First, probability has never been my strong suit, although I enjoy it quite a bit, I'm more interested in counting things, which is only a small part of that world. Second, we've left out some possible assumptions, e.g. symmetry of spawn points (there are two pairs which are practically the same), aka pairs of spawn points are indistinguishable. Third, my brain is turned off for the day

Any fellow mathematicians/actuaries feel free to correct my work. I'm too lazy to re read at this point, but if you PM me (to avoid a possibly lengthy back and forth) with an error or better way, I'll be sure to credit you in an edit (or I guess just post it of course).

 

This was extremely informative, thank you! Hopefully this further facilitates the necessity to remove the randomization of bonus spawns, or at least highlights the several inconsistencies (and therefore issues) that the current implementation causes.

 

This was brought up prior to Valentine's in hopes to get a definite answer, but it seems that things will be kept the same, mainly for consistency. The majority vote in this thread (one person per spawn point) is simply too OP. One spawn point for the whole team may be a good middle ground, but our internal discussion is too split.

Perhaps it will be discussed again at a future date. Moving this to closed for now.

 

So all this discussion for almost 1 year was worth nothing?
That was a very productive discussion.