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.