Guides/Probability

Probability guide

This page intends to provide some basic information on how the mathematics behind Eternal Lands basically work, what probability certain event have, say make an enriched essence in a certain number of essences, and what to expect when playing. We aim to skip the unnecessary stuff and present the results with some hopefully helpful examples.

Basic considerations

There underlying problem in determining probabilities and related values is classically an Urn problem: Let's say, the chance for an item to break is 1 in 300 (uses, hits). Imagine an urn with 299 white marbles and 1 black marble. Take a marble from the urn, look at it and put it back in. Now, the chance to pick the black marble is obviously 1 in 300, thus the urn models our problem.

The Eternal Lands server does exactly the same thing as picking up a marble, looking at it and putting in back when determining if your item breaks during this use/hit. Of course, the server doesn't need the visual aid, a 5% chance (5 in 100 equals 1 in 20) is computed like this

if(my_rand(20)==2)

according to Entropy, this checks if a randomly generated number equals 2 (the black marble).

Now, you'll probably want to use your items more than once which leads us to the next section:

Trying fate n times

First off: n stands for a natural number, such as 1,2,3,4,... and so on. Another common abbreviation in probability theory is to denote the chance of success with p and the chance to fail (no success) with q, obviously

q = 1 - p.

When you think of break rates you might not want to call your items breaking a success, so just think of p as the chance for an event to happen instead. The important part is that there's only two possible outcomes like in the example with the marbles above.

Now imagine that you're repeating the example above with the marbles a couple of times, say 3 times of picking up a marble, looking at it and putting it back. The chance to draw the black marble is 1 in 300 (p=1/300, q=299/300), you can get the following results (w=white, b=black):

(w,w,w), (w,w,b), (w,b,w), (b,w,w), (w,b,b), (b,w,b), (b,b,w), (b,b,b).

Obviously we'd expect (w,w,w) most of the time with a probability of (299/300) to the power of 3, but this highlights another problem:

When looking at break rates, do we want to know the probability for our items to break at a certain time or do we want to know how many spare items to buy or bring? Now, the first option certainly has interesting applications (say 2 EFEs right after another?) but mostly we'll be concerned with the second option.

I don't care when but k in n

where k is another natural number, less or equal to n. In the example above we find 3 possible ways to get a series of colored marbles with one black and two white marbles (we don't really get to see it as we have to put the marbles back), each of them has the chance of (299/300)(299/300)(1/300), so the probability to get one of those series is 3 times that. Now remember, we had p = 1/300 and q = 299/300, so we can write the probability as

3 * p^1 * q^2.

The general case, to make a long story short, looks like this

( (n!)/(k!(n-k)!) ) * p^k q^(n-k),

where the first term is the binomial coefficient of n and k.

We can now use the above formula to calculate what to expect in theory. In probability theory there's a so called expected value, that tells us what we should expect from an experiment "on average". The expected value while rolling a regular die is 3.5 by the way, even though you will never actually roll a 3.5 on a regular die. The expected value of the above experiment is

n*p.

In our example with the marbles this means that if we made a new experiment in which we repeated the old experiment 300 times (series of 300 marbles), we can expect to see the black marble once on average in a series. In other words: if you have an item that has a chance to break of 1 in 10k, you should expect it to break in 10k uses/hits.

There's one more value to consider here and that's the variance: it tells us how far spread out we should expect results from the expected value, think of it as a measurement to tell the extremely lucky/unlucky from the regular lucky/unlucky people. For our problem the variance is calculated to

n * p * q.

Putting this to use in our series of 300 marbles example we see that the variance is q (as n*p=1), which means we shouldn't really be expecting to have series with more than 2 black marbles in them (0 is possible of course). However if we did series of 600, we'd expect 2 black marbles per series on average and as much as 4 and as low as 0.

Other considerations

Keep in mind that the server always adds a small random number to your chances, there is no level where p=1 for anything except for the Magic skill, for all other skills, you will always have to work/calculate with a p<1.