Talk:Carefree Melody/@comment-38252036-20190121180758/@comment-38252036-20190122070340

I looked around for a number of figures to use but didn't see a convention in the guide, so I just posted the whole number so that it'd at least be in the edit history for other people to use. Since I've got a whole post's worth of space to use though, here's an (abridged) explanation for passerby:

Suppose we continue taking trials until the effect goes off 12500 times on the 8th hit or earlier. The 1st hit has a 0% chance, and thus will have a pass/proc ratio of 125000:0. The next hit has a 10% chance, and thus will have a ratio of 90 vs 10% of the previous passes, which is 112500:12500. The next hit has a 20% chance, and thus will have a ratio of 80 vs 20% of the previous passes, which is 90000:22500. The next hit has a 30% chance, and thus will have a ratio of 70 vs 30% of the previous passes, which is 63000:27000. The next hit has a 50% chance, and thus will have a ratio of 50 vs 50% of the previous passes, which is 31500:31500. The next hit has a 70% chance, and thus will have a ratio of 30 vs 70% of the previous passes, which is 9450:22050. The next hit has an 80% chance, and thus will have a ratio of 20 vs 80% of the previous passes, which is 1890:7560. The next hit has a 90% chance, and thus will have a ratio of 10 vs 90% of the previous passes, which is 189:1701. This leaves 189 attacks taken where the proc did not occur on the 8th hit or earlier, but they don’t need to be taken as we know 90% would have occurred on the next hit, 90% of the remainder would have occurred the hit after, etc (technically I think going up to the 8th hit was unnecessary, but I wanted an integer there).

Adding up both sides of each ratio gives us a total of 558340 attempts at taking damage, of which 0 were on the first turn, 12500 were on the second, 22500 were on the third, 27000 were on the 4th, 31500 were on the 5th, 22050 were on the 6th, 7560 were on the 7th, and 1701 were on the 8th; this means 0% were on the first, 10% were on the second, 18% were on the 3rd, 21.6% were on the 4th, 25.2% were on the 5th, 17.64% were on the 6th, 6.048% were on the 7th, and 1.3608% were on the 8th specifically, with a tenth of that happening on the next hit, a tenth of that on the one after, and so on, adding up to a joint 1.512% to occur on hit 8th or later (100% minus probabilities 1 through 7—you don’t actually need any calc to figure out the sum). This the same as the probability of having the charm NOT activate BEFORE the 8th turn, which means congrats; if you’re trying to get it up to 90% before a fight, you’ll have to activate the charm 66 times on average before that happens.

As for finding the average, I just iterated the first process up to the 30th hit for non-integer values and then took an average. Probably should have used calc, but 30 is overkill enough considering the 9th hit has an 18.9:170.1 ratio, 10th 1.89:17.01, 11th 0.189:1.701… 12th 0.0189:0.1701… they get small pretty fast.