Distributions of inclination to choose the first of two options I am trying to build a toy model of school student misbehaviour to compare with data I've taken from my classes.
A element of the models I'm considering is, when students choose an action, they may either choose to follow rules or break them. To begin with, each student has a probability to break the rules when they make a choice, between 0 and 1.
I want to know what sorts of distributions to consider for the students' probability to break rules. Ideally, it would be concentrated near 0 (most students follow rules).
My first thought is to try a truncated Gaussian, but my only reason for this is because it's such a common distribution.
 A: Clearly you're dealing with continuous probabilities on (0,1). There's an infinite number of distributions that fit that. (Truncated Gaussian can work if it does what you need, but it's not as easy to work with as a number of other options.)
Possibly the most common choice for such a distribution is the beta distribution.
It has two parameters and can have a variety of shapes. It can look close to normal, or it can be right or left skew, it can have a "U" shape, and so on:

We have:

*

*uniform (black)


*symmetric about 0.5, hill-shaped (green)


*reasonably right skew (blue)


*slightly left skew hill-shape (dark red / brownish)


*U shaped but asymmetric (magenta / pinkish)
... and I'm not even really sure what the best way to describe the red one is.
The distribution may have a wide spread (as the above ones all do) or it can be very tightly concentrated about some point.
You say you want to have something that can be concentrated near 0.
To get a beta distribution to concentrate near zero, the second parameter (beta) should be much larger than the first parameter (alpha), and should generally exceed 1 (or you'll get some concentration near 1 as well).
If you want the values near zero to have a peak somewhat above 0, alpha should exceed 1. If you want the peak at zero, alpha should be less than or equal to 1.

It's easy to make the densities concentrate even more strongly near 0 if desired.
However, you should not restrict yourself to this family if it doesn't match what you think you are dealing with. For example, if you wanted a distribution that had two peaks with at least one not at 0 or 1 then it would not be suitable (but a mixture of beta densities might do well).
Without more information I hesitate to leap into suggesting more distributions - if this suggestion doesn't include what you want you'll need to give more details.
