Drop chance probability
by Brian Wood 
Jan 13th 2010 at 11:00AM
If you hope to get the rare Deathcharger's Reins mount from Baron Rivendare for example, we know that it has a 1% drop chance. That means that every time you kill the Black Baron you have a 1% chance of getting the mount. 1% on the first kill, and 1% on the 100th kill. However, over the course of 100 kills, you have a much higher probability of getting the mount. But not 100%. Never 100%.
Join me after the cut where we take a friendly and gentle look at understanding probability, and give you a cool tool to automatically calculate drop probabilities for you!
Your Chance on Each Attempt is the Same
If your mount has a 1% chance of dropping, it will always be 1% on every single attempt. The Gambler's Fallacy trap that many fall into is assuming that previous results will change future results -- or put in WoW terms, "I've run this 50 times, so it must be really likely to drop now!" And it really feels like it should be that way. But it's not any more likely on the 50th kill or the 500th kill, it's still 1%. Every. Single. Time.
As an example, let's say you're flipping a coin. There's a 50% chance that it lands heads. You flip and get tails. Next time the chance is still 50%. Tails again. The next chance is still 50%. The past results don't affect the chance of the next result. That's randomness for you.
Your Chance Over Multiple Attempts Increases
Even though your chance on each single attempt always remains the same, the probability of getting your drop over the course of multiple attempts increases. I know at first this sounds like crazy talk that contradicts what we just discussed, so another example is in order:
We're flipping our coin again. We know that we have a 50% chance of getting heads on any given toss, and it doesn't matter at all what results we got before. But I think we can all agree that if we flip a coin 100 times it's very, very likely that we'll get heads at least one of those times. The chance on the first toss is 50%, and on the 42nd toss it's 50%, and on the 100th toss it's 50%. But over the course of 100 tosses, the probability of getting heads is way more than 50%. (In fact, the chance is 99.999999999999999999999999999921% that we'll get heads at least once.)
So the more often we down a boss, the more likely we are to see the loot that we want. Instinctively we all know this, that's why we keep going back and keep going back, and eventually our persistence is rewarded. Sometimes you'll get lucky, and sometimes you'll get unlucky, and the more you try the better your odds are overall. But the chance will never be 100%. It's never guaranteed.
How to Calculate it for Yourself
The formula to calculate your drop chance ( x ) over any given number of runs ( y ) is this:
1 - ( ( 1 - x ) ^ y )
Thus we can learn that if we kill Baron Rivendare 100 times, we have a 63.4% chance of getting the mount drop at least once.
Drop Chance Probability Calculator
But why do math yourself when the internet can do it for you! Just plug your drop chance and number of attempts into this tool to figure out the probability of seeing your drop. You can find estimates of the drop chances of most items on Wowhead, and be sure to check the comments to see if better drop chance info has been found than the estimates.
| Drop chance (enter 1% as 1): Number of runs: |
*Disclaimer: this script uses some rounding after lots of decimal places, so when very large numbers are input it can give you a 100% probability due to rounding errors. Your chance is never 100% unless the drop chance is 100%, and if you get that result, you can assume it's very, very, very close to 100%, but never quite there! Obviously entering negative numbers will get you silly results.
Reader Comments (Page 1 of 9)
siimmart Jan 13th 2010 11:02AM
what a nice calculator, /cheer!
Tebla Jan 13th 2010 1:11PM
I would rather see, let's say y (number of runs it will take) to have an acceptable probability (z) of getting the item with x drop rate.
For yucks, I tried to work that out myself. I had to look at log math again; it's been a while. This is what I got.
y = (log(1-z)/log(1-x))
Let's say I wanted a 75% chance of the reins dropping I would need to run Stratt 137 times. Dang.
Good thing it only took me 42. Seriously. 42. Weird, right?
UndeadReaperz Jan 13th 2010 8:01PM
Put into the calculater. 1 and then 1673. You get 100% chance on drop :) Happy Hunting - Only 1673 for Strath runs to go untill i get my mount!
Fatamorgana Jan 13th 2010 11:07AM
Anyone else miss Magic Find from Diablo...just a little bit?
Johan Jan 13th 2010 12:44PM
I remember the magic find sorc i had, dunno about the numbers, but it was pretty much. Kinda glad blizz scrapped that idea though :D
Drakkenfyre Jan 13th 2010 1:18PM
Math was hard for some of those players, too.
"I got 100% Magic Find! I have guaranteed Magic drops!"
"No, you have 100% increased chance to find. From normal. So you had a 1% chance, now you have a 2% chance."
Fatamorgana Jan 13th 2010 1:35PM
For nostalgia:
http://classic.battle.net/diablo2exp/items/magic/magicfind.shtml
I never broke the 1000mf mark, but was happy at 750. What I lacked in MF I countered with faster runs. Specifically Hell Meph runs in under two minutes on a meteor sorc.
Bedessa Jan 13th 2010 3:06PM
Mephisto was such a loot pinata. Oh the good days. I miss my immortal assassin. Phoenix strike was so win.
stylewalker Jan 13th 2010 11:11AM
All wrong, it's always 50% for everything to drop.
Either it does, or it does not, 50/50.
redrum Jan 13th 2010 11:19AM
That's like saying you have a 50% to roll a six on a six-sided die because you will either roll one or you won't every throw.
Blizzard clearly uses some sort of roll system to determine which loot drops. An internal roll happens and depending on the result different things drop.
if they use a 1-100 roll and you only get the mount on a roll of 4 it is a 1% drop rate.
Barinthos Jan 13th 2010 11:22AM
I took his comment as sarcastic jest... Either you get the item you want, or you don't. It's not REALLY 50/50.
Lynkfox Jan 13th 2010 11:24AM
Lets use the coin example again, but reverse it... What are the chances of you flipping a coin a hundred times and getting tails every time?
Everytime you flip the coin, the chance of getting the same side again is reduced. Yes, its still a 50% chance of getting either heads or tails for that individual flip but after taking the whole thing into account, it has less chance of flipping the same thing over and over again.
Therefore, the chance of getting the other side, actually increases, because the probability of flipping a coin a hundred times and getting 100 tails is vanishingly small. Even though, taking into account the individual toss, the chance is still just 50%
The same thing applies to 1% drops. Its a bit more complicated because of the number of other items that can drop, all of which have a greater chance of dropping than 1% (not the equal chances of head vs tails) but the basic theory is the same.
Protocollie Jan 13th 2010 11:26AM
@lynkfox - You're looking at it wrong. On a per-attempt basis there is no increase in probability. When you run once, the chance that the next run will drop the item is the same as the chance before.
When you're looking at a set of 100 runs however, the probability that that set of 100 runs contains the drop somewhere is significantly larger. On an instance-by-instance basis however the probability does not change.
Lynkfox Jan 13th 2010 11:33AM
@protocollie
Um... that is what i just said? That individually the drop is the same, but the chance that you will get all one side over a 100 is less...
Sagretti Jan 13th 2010 11:34AM
I'm pretty sure it's a sarcastic comment. Reminds me of, and may be a reference to, a Chuck Klosterman essay where Yoda's "Do or Do Not" becomes the driving construction of the universe. The chance of anything happening is 50/50, because there's only 2 possible outcomes, it does happen or doesn't. So, there is a 50% chance space aliens will blow up our planet at 3:15, because it will either happen or it won't. 50/50.
Norte Jan 13th 2010 11:37AM
@LynkFox:
"Everytime you flip the coin, the chance of getting the same side again is reduced."
No. The chance of flipping the same side as you just flipped is *always* 50%. Even if you just flipped ten heads, the chance of the next one being a head is 50%.
"No computer generated number is truly random"
Don't modern hardware implementations of random number generation take into account the heat of the cpu? For all intents and purposes, that's entirely random.
Henry Jan 13th 2010 11:41AM
Math Troll is a Math Troll
Mutak Jan 13th 2010 11:55AM
@Lynkfox - i believe you do understand the math correctly, but your terminology is wrong and could cause confusion for someone who does not understand it.
It is always incorrect say the probability increases or decreases per event. You can say that the probability of something to have happened increases or decreases.
The probability of Deathcharger's Reigns dropping is always 1%. The probability that it will have dropped increases as the number of chances increases.
Raioul of Shadow Council Jan 14th 2010 4:03AM
This isn't Schrodinger's Cat dude.
Jafari Jan 13th 2010 12:49PM
reply to Lynkfox
You say that there is a very small chance to get 100 heads in a row, therefore the probablity that the 100th throw after 99 heads is also heads is very small.
The problem with what you are saying is that the PREMISE, that 99 heads in a row have been thrown, is extremely unlikely. In fact, throwing 99 heads in a row and then a tails on the 100th throw is just as unlikely as throwing 100 heads.
Indeed, if you make a list of 100 predicted outcomes like this: HTHHHTHTHHHTHTHTHTHTHTHTHTHTHTHTHHHHHHTHHHTHTTTTTHTHTHTTTHHTTTTHHTTH(to 100)
then you are as likely to get that exact string of results as you are to get all heads, or all tails.
The likelihood of a prediction only changes if you are indifferent as to the order in which the results occur. For example, if you predict that on 10 flips you will get at least one heads, then you do have a higher than 50% chance to get it right, but if you predict that any PARTICULAR flip will be heads, it is 50% likely that you are wrong, regardless of the results of the other flips.