In mathematics, probability is conventionally expressed as a number between zero (no chance of happening) and one (guaranteed to happen). For example, if there’s a 14% chance the boss will drop your shoulders, we say the probability is 0.14.

The probability of something *not* happening is one, minus the probability of it happening. So the chance the boss will *not* drop your shoulders on any given kill is 1 – 0.14 = 0.86.

Now suppose (for simpler arithmetic) the item has a 25% drop rate, which is a probability of 0.25, or one-in-four. The probability of the item dropping on the first attempt (regardless of any subsequent attempts) is 1/4. The probability of it dropping on the second attempt (regardless of the first or any later attempts) is *also* one-in-four.

What’s the probability that it drops *both* times? Well, one out of four times, it drops the first time. But then, only one out of four of the *first* one out of four times, it drops the second time as well. One out of four *of* one out of four is 1/4 x 1/4 = 1/16. So, if you describe one “attempt” as killing the boss twice, only one in sixteen attempts will the boss drop the item on both kills.

In general, if X is the chance of one event happening, and Y is the chance of another event happening, and the events are *independent* of each other (in other words, the second event is not made more or less likely by whether the first event happened or not), then the chance of *both* events happening is X times Y.

Now we’re ready to look at drop rates.

If I run the dungeon 3 times, and the item has a 1/8 chance to drop every run, what I want to know the chance that it drops *at least once*. In order to do that, we calculate the chance that it *never* drops in *any* of the three runs, and subtract that from 1.

Since the item has a 1/8 chance to drop each time, it has a 7/8 chance to *not* drop each time. So the chances of it not dropping in any of the three times is 7/8 x 7/8 x 7/8:

0.875 x 0.875 x 0.875 = 0.669921875

So there’s a nearly 67% chance that the item does *not* drop in those three attempts. That leaves a roughly 33% chance that it *does* drop. So by repeating a 1/8 chance three times, it happens that we’ve gotten to a roughly 1/3 chance that we’ll get the item.

In general, if X is the probability of the item dropping in a single attempt, and N is the number of attempts, the probability P of getting the item in at least one of the attempts is:

P = 1 – [(1 – X)^N]

Example: Wowhead says the chance of getting Kael’thas’s Swift White Hawkstrider is 3%. So the chance of getting it in 50 attempts is:

P = 1 – [(1 – 0.03)^50] = 1 – [0.97^50] = 1 – [0.218…] ≈ 0.782 = 78.2%.

You can run Magister’s Terrace 50 times and there’s still a slightly better than one-in-five chance you’ve not seen it drop. Hopefully you were running it solo.