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Probability Expert?


Staff member
Anyone know probabilities? I'm trying to figure out whether a site is giving the correct information on the chances of getting a specific "item" from a "chest". Basically, the chances to get it seem to go up the more chests you open, which seems odd to me because opening a chest is an "independent event" (in probability terms) so it shouldn't matter whether you open one chest or 1000, the chances of getting the item should be the same no matter what.


The Tentacled One
I remember that BigBlue was a math person. Forget if we had anyone else...

The figure they're giving you may be for opening up all of those chests. Each one is independent, but we can still compute the probability of getting the desired item from a set number of chests.

Let's say that we have a simple scenario. There's 1 chest and it has a 50% chance of containing the type of item we want. Well, then our probability of getting the item is 0.5, or 50%. But let's say we have 2 chests and each of them has that same chance. What it means for those to be independent events is this: once we've opened up the first chest, the outcome has no effect on the probability of getting an item in the second chest. It's the same as if we were flipping a coin or rolling a die. The fact that we flipped heads first doesn't mean we're more or less likely to flip heads on the next event.

But if all we care about is getting an item of the type we're looking for, how do we find the probability of that when there are multiple independent events? Consider our simple 2-chest scenario with an independent 0.5 probability of success for each chest. There are 4 total ways this could go...
  1. Win, Lose
  2. Lose, Win
  3. Win, Win
  4. Lose, Lose
Each of those is equally likely (we know that because each opening is an independent event). But in light of our requirement, the only one of these we actually care about is #4. We specified that we want that damn item, and it doesn't matter to us when in the series of openings it pops up. It doesn't matter how many times it pops up. Just one occurrence is enough. So outcomes 1, 2, and 3 are, for our purposes, identical. So it's easy to see that our probability of failure, of not getting an item in the set of 2 chests, is 0.25 or 25%. And from that it follows that the probability of winning, of those 3 possible good outcomes combined, is 75% because 1 - 0.25 = 0.75.

It turns out we can extend this to larger numbers of chests. We don't need to map out all possible patterns of failed and succeeded openings. We just need to know the probability of getting all failures and then we subtract that probability from 1 to find our probability of any successful outcome.

Suppose we have some smaller probability of getting the item we want per chest. Let's say it's only 5%. That's 0.05. So the probability of not getting the item is 0.95 per chest. 0.95 × 0.95 = 0.9025. We have a 90.25% chance of both chests coming up empty, which means the chances of success are...

1 - 0.9025 = 0.0975 = 9.75%.

Now let's try it with 3 chests (using that same 5% success rate). 0.95 × 0.95 × 0.95 = 0.857375. 1 - 0.857375 = 0.142625 = 14.2625%.

With 4 chests? 0.95^4 = 0.81450625. 1 - 0.81450625 = 0.18549375 = 18.549375%.

5 chests? 1 - (0.95^5) = 0.2262190625 = 22.62190625%

And so on...
6 chests: 0.264908109375
7 chests: 0.30166270390625
8 chests: 0.3365795687109375
9 chests: 0.369750590275390625
10 chests: 0.40126306076162109375

But maybe I really want that item. So I'm going to open up 100 chests...
1 - (0.95^100) = 0.9940794707796659745170750351176. About 99.4%.

Does that help?


Staff member
That helps wonderfully! It's when you "mapped" out the four possibilities and explained it is where it clicked and made sense. So the game that the site is based on gives the option of opening 1 chest at a time, 50 chests at a time, and 200 chests at a time, and although it doesn't make a difference whether I open 1 chest 200 times or do the 200 chest option, because the *total* amount of chests that I'm opening is still 200, the chances of getting the specific item are increased overall by that much.

Uh... right? :D


The Tentacled One
Right! For independent events, we don't care in which order the chests are opened or how long it takes. We could say that we'll open them up one at a time or that we'll open them all up at once or somewhere in between. If we're still at a point before that has happened and we want to know the probability, the answer's going to be the same.

From what I remember of the statistics I took in school, the error in thinking that plagues some students is going past that point, revealing more information, and then treating it as though it's still that same original probability. For instance, let's say that I've got 200 chests and the probability of getting the desired item in each chest is, independently, 0.5%. A quick calculation shows that the probability of getting my item in there somewhere is 1 - (0.995^200) ≈ 63.3%. So it's in the realm of not really being close to a sure thing, but still pretty likely. Now, someone who is thinking things through properly and who knows that the starting probability is 63.3% would realize that as you go through the chests one-by-one and keep coming up with failures, it becomes increasingly likely that you're not in your desired 63.3% chance of success, but rather that you happen to have landed in the undesirable 36.7% part of the map. Once you've opened up 199 chests, it's easy to see that your probability of winning is not 63.3%, but instead is exactly 0.5% (which is extremely bad, albeit still not completely impossible). And it shouldn't be hard to realize that at every point along the way, our calculated probability would have been some number between 0.5% and 63.3%. Our chances steadily dropped as the failures racked up. It's a version of the "Gambler's fallacy" to hold onto that original 63.3% from when we had less information and to think, "We had a better than even chance when we started and we've come up dry so far, and that means success is just around the corner." And of course it doesn't work that way. Like you said in your first post, these are independent events. Failing to score 199 times in a row doesn't make that 200th chest any more or less likely to give your item. The difference between opening up 200 all at the same time and failing vs. opening them up one-at-a-time and failing is that the latter is slower and more agonizing. The end result is the same and is, from our starting point, equally likely.


Staff member
Heh, yeah. I like opening them at one at a time for x times "just in case" I get it, in which case I saved 200-x "keys" for the next go-around (once a month). But it is frustrating to open 150+ plus and not get it and think "man, I should just click the Open 200 button and I'll be sure to get it then".