User blog comment:BunsenH/Ghazt Stats Phase 3: Wishing Torches - Report 2/@comment-26245311-20150326004904

I'm a little rusty on my statistics (specifically with conditional probabilities). But I'm comparing "Report 1" and "Report 2", both from Ghzat Stats Phase 3. And it seems you basically have your answer already.

On Report 1, there were 47 successful of 2002 total. On Report 2, there were 77 successful of 4004 total. Now, I assume Report 2 is "total since the start", not "total only since posting Report 1". If my assumption is correct, you/minions got really lucky in the first 2002 attempts, getting 47, but then got very unlucky in the next 2002 attempts, getting only 30 in the same number of attempts.

The more interesting bit is the probabilities you've listed, including their accuracy. Because of the great variation of luck from the first 2002 attempts to the next 2002, I think you've already found your answer. Report 1 shows 2.35+/-0.34% (meaning the value could be from 2.01 to 2.69%). Report 2 shows 1.92+/-0.22% (meaning the value could be from 1.70 to 2.14%).

Look at those 2 logically, that means the probability lies somewhere between 2.01 and 2.14%. So even though your full sample of 4004 has a greater variance, it's been narrowed down by looking at both portions of the sample. (I'd be interested to see what probability/variance you'd come up with for the 2003-4004 samples, which provided 30 of 2002 attempts = 1.50+/-???%, to see if it could narrow that down even a slight bit more?)

I'm wondering if there might be a bit of rounding in your 2.35+/-0.34 (or if you were literally on the maximum edge of your 95th or 99th percentile or whatever you're using)... but it basically looks to me like no torches is 1.00% probability, and 1 perma-torch is 2.00%, and that torches increase ghzat chances by 1% each [effectively the probability is (n+1)*(base)], but that wouldn't transfer to higher-probability breeding monsters (a 15% chance monster, for example, with 10 torches, would have 165% chance of successful breeding, which is not logical).

So maybe a torch is 1%, regardless? On a 1% monster, 10 torches can bring you from 1/100 base to 1/9 chances (11%). And the effect on a 50% monster is less noticeable, since it would only be up to 60%.

The other thing I thought of... maybe it's not linear, but exponential. The chance of breeding with 1 torch is [(1+base)^(n+1)-1], so for ghzat, assuming base is 0.01, that's 1.01^2-1 which is 0.0201 or 2.01%, and that would fall within the "Report 1" findings listed above (which is pretty much the reason I though it might be exponential instead of linear). However, this logic doesn't transfer to a 50% breeding monster, for example, where with even 1 torch lit, 1.5^2-1=1.25 or 125%... same we "transferability" as above.

So the way around that is that the probability of "not breeding successfully" could be exponential. [1-(1-base)^(n+1)] ... look at it with ghzat's, assuming 1.00% base probability. 1 torch brings that to 1.99% (slightly beyond "Report 1" findings, but awefully close). 10 torches would bring it to 10.5%, which would be fairly close to the 11% if each torch was exactly 1%. So if you do a test with 2 or 3 torches, even getting thousands of results will be hard to narrow down the variance enough to distinguish it, since the difference at 10 torches is still very small between these two options.

How do you get around that? Test it on a higher variation monster, as you've noted. Maybe do the rest of the wishing torch tests on breeding an Entbrat, as you mentioned. If it's 10% (I have no idea, but I'll use this number for comparison)... linear 1% per torch would increase it to 20%. Exponential chances of not breeding successfully would be 1-0.9^11=0.686 or 68.6% chance of a successful breeding. That is a huge enough difference that a few days of testing 10 torches (or a few weeks of testing with 2 or 3) would definitely give you an answer, whereas testing even 10 torches on a ghzat could take months to narrow down the variance enough to distinguish between linear and exponential.

Linear would be the simplest for coding, and would seem to make sense. The only exception might be if there were a 91-99% chance monster - what would happen if it had 10 torches lit? But I guess it could just max out at 100%.

However, even having said all that, I understand the need to be thorough. So continue your testing routine, and list results, and I will keep reading them (even if it's infrequently, judging by the length of time between when you posted Result 2 and when I replied). I just thought I'd share my thoughts, and see if there was a way to speed up results on some of the testing.