User:Yamagawa/Zero Drops

The 0 Drops Problem
The problem:

We have a new item in game, the Drops-A-Lot Event Bag.

It's been announced that this bag drops a uber-cool widget, the 'Green Eyes of Envy'.

Everyone wants a pair.

So you go out and open some Drops-A-Lot bags.

You open 5 bags, and get no Eyes. Surely, they must occur less often than 99% of the time.

You open 50 bags, and get no Eyes. Surely, they must occur less often than 25% of the time.

You open 1000 bags, and get no Eyes. Surely, they must occur less often than 5% of the time...

But how often is that? Can we get a concrete number for that (with the classic 'odds are 95% certain that...')?

I say we can.

Lets take a rate, say, 10%... and decide how many bags we would need to open if it dropped at that rate, to be 95% certain to get the eyes.

r = rate n = # of opens. o = odds. o = 1 - ( ( 1 - r ) ^ n ) o - 1 = - ( ( 1 - r ) ^ n ) 1 - o = ( 1 - r ) ^ n n = LOG(base 1-r of 1-o) n = 28.43315881 So at a 10% drop rate, we'd need to open 28.43315881 bags to get our eyes of envy 95% of the time. But, does that mean that if we open 28.43315881 bags and don't get the eyes, the drop rate is 95% certain to be lower?

No!

The 95% certainty listed above is the chance of GETTING what you want 95% of the time. That is different form a 95% chance of the rate being below 10%, if you open n bags. The one applies the 95% to your hopes and prayers of getting what you envy. The other applies the 95% to the probable rate of occurrence. To do: Determine what formulae expresses with 95% certainty, that n no drops means the rate is below r.