curved (or angled) probability

Question

In nearly any programming language, if I do $number = rand(1,100) then I have created a flat probability, in which each number has a 1% chance of coming up.

What if I'm trying to abstract something weird, like launching rockets into space, so I want a curved (or angled) probability chart. But I don't want a "stepped" chart. (important: I'm not a math nerd, so there are probably terms or concepts that I'm completely skipping or ignorant of!) An angled chart is fine though.

So, if I wanted a probability that gave results of 1 through 100... 1 would be the most common result. 2 the next most common. In a straight line until a certain point - lets say 50, then the chart angles, and the probability of rolling 51 is less than that of rolling 49. Then it angles again at 75, so the probability of getting a result above 75 is not simply 25%, but instead is some incredibly smaller number, depending on the chart, perhaps only 10% or 5% or so.

Does this question make any sense? I'd specifically like to see how this can be done in PHP, but I wager the required logic will be rather portable.

Answer 1

The short answers to your questions are, yes this makes sense, and yes it is possible.

The technical term for what you're talking about is a probability density function. Intuitively, it's just what it sounds like: It is a function that tells you, if you draw random samples, how densely those samples will cluster (and what those clusters look like.) What you identify as a "flat" function is also called a uniform density; another very common one often built into standard libraries is a "normal" or Gaussian distribution. You've seen it, it's also called a bell curve distribution.

But subject to some limitations, you can have any distribution you like, and it's relatively straightforward to build one from the other.

That's the good news. The bad news is that it's math nerd territory. The ideas behind probability density functions are pretty intuitive and easy to understand, but the full power of working with them is only unlocked with a little bit of calculus. For instance, one of the limitations on your function is that the total probability has to be unity, which is the same as saying that the area under your curve needs to be exactly one. In the exact case you describe, the function is all straight lines, so you don't strictly need calculus to help you with that constraint... but in the general case, you really do.

Two good terms to look for are "Transformation methods" (there are several) and "rejection sampling." The basic idea behind rejection sampling is that you have a function you can use (in this case, your uniform distribution) and a function you want. You use the uniform distribution to make a bunch of points (x,y), and then use your desired function as a test vs the y coordinate to accept or reject the x coordinates.

That makes almost no sense without pictures, though, and unfortunately, all the best ways to talk about this are calculus based. The link below has a pretty good description and pretty good illustrations.

http://www.stats.bris.ac.uk/~manpw/teaching/folien2.pdf

Answer 2

The easiest way to achieve a curved probability is to think how you want to distribute for example a prize in a game across many winners and loosers. To simplify your example I take 16 players and 4 prizes. Then I make an array with a symbol of the prize (1,2,2,3,3,3,3,3,4,4,4,4,4,4,4) and pick randomly a number out of this array. Mathematically you would have a probability for prize 1 = 1:16, for prize 2 3:16, for prize 3 5:16 and for prize 4 7:16.

Answer 3

To me it seems like you need a logarithmic function (which is curved). You'd still pull a random number, but the value that you'd get would be closer to 1 than 100 most of the time. So I guess this could work:

function random_value($min=0, $max=100) {
    return log(rand($min, $max), 10) * 10;
}

However you may want to look into it yourself to make sure.

Answer 4

Essentially you need only to pick a random number and then feed into a function, probably exponential, to pick the number.

Figuring out how weighted you want the results to be will make the formula you use different.

Assuming PHP has a random double function, I'm going to call it random.

$num = 100 * pow(random(), 2);

This will cause the random number to multiply by itself twice, and since it returns a number between 0 and 1, it will get smaller, thus increasing the chance to be a lower number. To get the exact ratio you'd just have to play with this format.