Hey great post, the math is all very well written and I really enjoyed reading it. I was actually considering making a maushold post cause the math is so interesting but we banned king's rock so quick that I scrapped the idea. Regardless, this now gives me a good an excuse to make a post about math in the meta discussion thread. About 2 weeks ago I used population bomb with kings rock 100 times and wrote down all the data within a spreadsheet, for an idea of what the probability actually look like. Out of 100 hits, population bomb flinched 43 times, and got an average of 5.7 hits per use. Here's the spreadsheet if you wanna take a peek at the data: https://docs.google.com/spreadsheets/d/1jss82VdBCnrnwn92anfly_Gosiq-E4iaNrmJEE9tNaA/edit?usp=sharingQuick correction on population bomb with king's rock.
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Population bomb can miss after each hit and king's rock's flinch chance will compound with each hit. So while 10 hits does have a 1-(0.9^10)= 65.1% chance of flinching the opponent, you'd need to factor in the fact that it will only land .9^10= 34.9% of the time.
Next, you need to factor in all the other outcomes as well such as landing 9 times all the way down to landing once and add them up.
This comes out to roughly 19.25% chance to flinch.
That said, I can predict that Maushold will get some way to boost accuracy through Hone Claws (or more disturbingly, coil), which would make King's Population Rock Bomb truly terrifying.
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I am still in favor of the ban as kings rock, bright powder, & razor fang could allow for the tiniest odds to flat out lose thanks to the item alone.Odds of King's Rock = 10%, Accuracy of Population Bomb per hit = 90%. (funny how they add to 100%)
Formula for a flinch per a hit = (Accuracy^Hits) * (1-Flinch Chance^Hits)
Odds of zero hits: (.9^0)*(1-.9^0)
Odds of a single hit: (.9^1)*(1-.9^1)
Odds for 3 hits: (.9^3)*(1-.9^3)
Odds for 10 hits: (.9^10)*(1-.9^10)
Average each outcome to get close to 19.25% flinch chance on average.
EDIT: first way was absolutely wrong, and there is nothing to give you motivation and a clear mind than putting your math out in front of everyone.
While your math and logic is fairly sound, there's just one inaccuracy that really causes some things to go wrong. Here's a comparison we can look at: by your calculations population bomb would have an 81% chance to hit twice and a ~35% chance to hit 10 times. This woul then imply that population bomb will hit 2 times more than twice as often as it hits 10 times. However the data contradicts this and there's a reason why. Once population bomb hits twice, it's not done hitting, and can go on to hit 3 times, 4 times, 5 times, etc., and the chance of it hitting just 2 times is slimmer when taking into account the chance to keep going. However, after population bomb hits 10 times, it's done checking the accuracy, and 10 is simply the stopping point. This gives population bomb a much higher chance of hitting 10 times, and this is shown in the data where population bomb got 10 hits 33 times out of 100, much more than I initially expected.
From this new insight we can actually figure out the population bomb average hit chance pretty easily. The chance to hit 0 times is 0.1 or 10%, which is the chance to just not hit the first one. The chance to hit 1 time is 0.09 or 9%, which is the chance to hit the first one(0.9) times the chance not to hit the second one(0.1). The chance to hit 2 times is 0.081, which is the chance to hit twice(0.81) times the chance not to hit the third one(0.1), and this goes on and on up to 9 hits. 10 hits has a 0.348 or 34.8% chance of occurring, which is just 0.9^10. Here's a picture of each of the hit chances:
Unfortunately, just like Chrispy Burns, my math knowledge is still limited and this could very well be wrong. I tried to use real game data to make up for any inaccuracies in my calculations though, and you can feel free to add to this conversation by correcting any of my assumptions.