Some remarks on drop rates

I'm going to keep this relatively short, because a full discussion of probability could fill several college semesters. However, there is one misconception that some WoW players have that has been bugging me lately.

Let's say you read that Shattered Sun Supplies have a 10% chance to contain a Badge of Justice, and, excited, you go out and do enough dailies get 10 Shattered Sun Supplies. You open them all and find not a single Badge, or you find five badges. Do either of these outcomes mean the 10% drop rate is wrong? No! They do not! All a 10% drop rate means is that for each Supplies, there is a 10% chance that it contains a Badge. Random events have no memory, so no matter how many badges you get in the first nine Supplies, your chance to get a Badge in the tenth Supplies is still 10%. The traditional analogy is that if you flip a coin nine times and get heads each time, the chance of getting heads on the next flip is still 50%.

Now it is true that you will probably get a Badge in ten Supplies if the drop rate is 10%. If you're interested in how likely it is, here's the calculation to do. The chance of not getting a Badge in one Supplies is (100% - 10%) = 90%, or 0.9. Raise that to the tenth power, for your ten independent Supplies-opening events, and you get the chance of, ten times out of ten, not getting a Badge: 0.9^10 = 0.349, about 35%. So in fact, out of ten Supplies, you will get a badge (100% - 35%) = 65% of the time, about two thirds.

TL;DR version: A drop rate is a probability, not a guarantee.