I don't really know how to describe my problem in maths terms, so maybe I try to describe what I like to calculate:
In video games, there are drop chances. When you kill a boss, there is an N% chance for a specific item to drop.
E.G:
Item A drops with a chance of 1%
Item B drops with a chance of 5%
Item C drops with a chance of 10%
I already found out how to calculate the chance for an item to drop after N amount of tries.
E.G:
Item A (1% drop chance) has a 63.4% chance of dropping at least once after 100 tries.
Item B (5% drop chance) has a 40.13% chance of dropping at least once after 10 tries.
Item C (10% drop chance) has a 87.84% chance of dropping at least once after 20 tries.
But now I'd like to know how likely it is combined for all those tries. In other words:
How likely is it, that after 100 tries at 1%, 10 tries at 5% and 20 tries at 10%, no item dropped yet?
Also, to add a bit more complexity. Is there a way to tell the luckiness or unluckiness when some items dropped, but some didn't yet.
E.G:
How lucky or unlucky (in percent) is it, when: Item A dropped after 100 tries (63.4%), item B didn't drop yet after 10 tries (40.13%), and item C dropped after 20 tries (87.84%).
Is there any way to calculate a meaningful likeliness that can tell if I am X% lucky or unlucky?
Best Answer
Assuming the drop probabilities of items A, B, and C are independent, you have a 1-0.01=0.99 chance of not getting item A, and similarly 0.95 chance of not getting item B and 0.90 chance of not getting item C, on any given try.
Then, you can simply raise these to the relevant powers.
For this question, the answer is $0.99^{100} \cdot 0.95^{10} 0.90^{20}$.
As far as I know, your notion of "luckiness" or "unluckiness" doesn't seem to be too clear, but I think the best way to look at it would probably be looking at where the number of tries falls on the binomial distribution.