A Guide to Probabilities
By Epiphenomenon
Many items give a certain probability of triggering a special ability. At first glance, many of these special abilities seem unhelpful. How often is a 5% chance really going to come in handy? Is a 1 in 20 chance really that helpful? Learning a bit about probabilities will help you determine whether an item or skill will be useful.
First, let’s define a few terms. First, we will assume that each chance-to-trigger is an independent event. This means that the odds of triggering an ability is not dependent on previous successes or other factors. Journeyman’s Treads have a 5% chance to trigger on being hit, and this never changes.
Now let’s consider the following scenario. The Rook’s poisoned arrows skill has a 15% chance to slow a target for 10% for 3 seconds. That seems really low. However, the Rook fires a sizeable volley of arrows almost simultaneously. How do we figure out the odds of triggering this special ability if five arrows hit an opponent simultaneously?
We do NOT merely add up the probabilities of each arrow’s chance-to-trigger and call it good. If we did this, we would get 0.15*5 = 0.75. This is not correct. If we used 10 arrows in this example, it we would get a probability of 1.5 for this method. Probabilities must always be between zero and one, so obviously, this would not work.
The easiest way to figure this out is to determine the odds that none of the arrows will hit, and then subtract our answer from one. This will give us the probability that 1, 2, 3, 4, or 5 arrows will hit. We don’t really care if more than one arrow hits, because they do not stack (as far as I can tell). All we want is at least one arrow in our volley to hit.
Since these events are independent, we simply multiply the probabilities together to figure out the odds of none of them hitting. The probability of one arrow not triggering “poison arrows” is 0.85. The probability of two arrows not triggering the event is 0.85*0.85. The probability of five arrows not triggering, then, is 0.85*0.85*0.85*0.85*0.85 = 0.444. We could also write this as 0.85^5. Probability encompasses all possibilities. So, if we subtract that number from the number one, we get the odds that at least one of them will hit. This gives us the probability of 1 – 0.444 = 0.556. We have a 55.6% chance of slowing our opponent if five measly arrows hit him! These are great odds, as the Rook fires a ton of arrows.
What about Journeyman’s Treads? It has a 5% chance to increase movement speed by 50% on being hit. Let’s say you get hit by one low-level group of minotaurs and archers once. Let’s assume you get hit simultaneously by the 5 minotaurs and the 2 archers. This gives us seven hits. The odds of none of them triggering Journeman’s Treads is therefore 0.95*0.95*0.95*0.95*0.95*0.95*0.95 (which can be written as 0.95^7). This number is 0.698. Our odds of triggering the event after being hit once are therefore 1 – 0.698 = 0.302. What if the creeps hit you twice? Our odds of none of them hitting you two times in a row is 0.698^2, which means our probability of triggering is 1 – (0.698^2) = 0.513. If we were to add in priests, catapultasauri, and giants, the odds of triggering Journeyman’s Treads would be much higher. You can do this on your own if you want. 
Short version:
1. Figure out the odds that a special ability will fail to trigger.
2. Raise this number to an exponent. The exponent is the number of times it is possible for an event to trigger. For example, if it is an item that triggers on being hit, and you get hit seven times, you would raise your probability to the seventh power.
3. Subtract this number from one. The answer is the odds that your special ability will trigger after being hit seven times.
[edit] Zechnophobe put it a good way. Here's the actual equation: 1 - (NOT^Chances)
Hopefully this was helpful. If you have any questions, feel free to ask.
