I used to think the number of spins/chest opens/whatever you did at a time didn't matter at all. The tricky thing is that whether it does or doesn't is really hard to prove, and it's easy to have your judgment clouded by confirmation bias (basically, you tend to only notice the things that are irregular, even though many things are normal, so one observation outside the scope of a planned experiment is not something to draw conclusions from). However I've observed a few things that are just too peculiar to ignore.
Let me just say one thing to start: I believe your expectation (that is, the number of [item] you would receive if you could open an infinite amount of chests) is the same no matter how many you open at a time. What's interesting is that the results don't appear to be normally distributed when you do things in clumps, as they would be if you did them one at a time. There appears to be less variance than expected when opening large amounts of things at once, meaning your results will be closer to the average more often. I suspect that, rather than sequentially perform X number of rolls as I had initially assumed happened, PK is programmed to estimate a reasonable outcome (to reduce server load), and the code to do this is a bit faulty. This may be the reason you are limited to opening 100 of any chest/bag at a time -- the results would be too "unrealistic" otherwise.
I wrote this page a couple of months ago after getting my first spin hero. I believe the probabilities are correct if you are performing 1 spin at a time. I've now spun 4 heroes and they have all taken between 8,400 and 8,800 tickets by doing 100x spins until I am within 3 pieces and then 10x for the rest. This would be a range of about 10% on that table, meaning one in ten spin attempts should fall into that range, and meaning it is very odd that four in a row have.
Two more peculiar things:
- I've done 31 100x spins. I've only gotten 4 pieces once. The odds of getting 4 or more pieces, with pieces dropping at 1/45 chance, in 100 spins, should be 18.9%. The chance of getting 4 or more pieces fewer than twice in 31 attempts is 1.008%
- I've also never gotten 0 pieces. The odds of getting 0 pieces in 100 spins should be 10.3%. I should expect to get 0 pieces 3 times in 31 100x attempts. The chance of that never happening, if 100x draws were done fairly, would be 3.44%
Ok, unusual but not wildly so, and it wouldn't be wise to draw any conclusions from that. Look at this though.
- In about two months of farming courage gems and hitting the cap 6 out of 7 days, I have never gotten two event chests from opening between 30 and 55 courage chests at a time. It is always either 1 (usually, and seemingly almost always with 50+ chests) or 0 (rarely -- more likely with fewer opens, naturally). If the odds of finding an event chest are 1 in 45 (educated guess), the odds of finding 2 or more event chests from a batch of 40 courage chests should be 22.7%. If I've opened one batch of 40 chests each day for 60 days (in reality, between 35-55 a day), the odds of never finding 2+ event chests over that time are one in 5.1 million.
But, like I said, I don't think your expectation changes at all. I have no idea how many times exactly I failed to get an event chest from opening a bunch of courage chests, I just know that it happens. It's possible that it happened a lot less frequently than it "should" have, balancing all the times I "should" have been lucky enough to draw multiple event chests.
The one conclusion I will make from this is that it is riskier to open chests and spin in smaller quantities. If you perform 10x spin, it is possible it may take over ten thousand vouchers to gather 20 pieces, but it is also possible that it takes under six thousand, whereas both these outcomes seem incredibly unlikely performing 100x spin.
So, it still doesn't matter how many spins you do at a time, just not for the reason I expected.