Probability Professor Pikken Ponders

I have done <x> 30 times and not once has a <y> dropped, RNG is broken!

Statements like the above we hear from time to time on boards like these.

Now let's calculate how unlikely this bad luck is. Let's assume we are talking about getting 1 specific drop out of 8 possible drops, like the Garuda weapon for example.

The chance to get the drop each time is 1/8 = 0.125
The chance to not have gotten the drop after x attemps is 1 - 0.125^x

So,
after 1 attempt, there is a 88% chance to not have gotten it
after 10 attempts, there is a 26% chance to not have gotten it
after 20 attempts, there is a 7% chance to not have gotten it
after 30 attempts, there is a 2% chance to not have gotten it

2% may seem like a small number. But seen in the context of how many people are involved, it really isn't. In every 100 persons, the Expected value for the number of people not having gotten their drop after 30 attemps is 2. Or 1 in 50.

Let's say 50 000 people are grinding this example, it is expected that about 1000 of these have failed to get their drop after 30 attempts. Adding to this there is most likely also a Publication bias here. People who have failed 30 times are more likely to post it on the boards than, say, someone how got the drop after 8 attempts.

And most importantly, it's never 30 attempts. The amount of attempts are always padded by a factor of 5 which is the SRSL SE?! constant.

There are actually 9 weapons (SMN/SCH are separate), so there is a 3% chance you won't get it after 30 attempts.