Probability distribution of halving a segment Suppose you cut a unit segment ("a stick") in half. I'd like to find out theoretically the distribution of the length of, say, the left piece based purely on some plausible assumptions, such as symmetry (i.e. the mean is $1/2$ and the distribution is symmetric around it) and that you are never going to completely miss the stick - the length is always positive (i.e. the distribution drops off to $0$ at $0$ and $1$).
I understand that this question might be somewhat open-ended based on what assumptions one considers plausible.

A family of distributions often used to model the breaking of a linear piece is the beta family. For every $a\gt0$ and $b\gt0$, the beta $(a,b)$ distribution has density 
$$
\mathrm B(a,b)^{-1}x^{a-1}(1-x)^{b-1}\mathbf 1_{0\lt x\lt1}.
$$
Some justifications for this choice come from Bayesian considerations. Several shapes are possible. For shapes dropping to zero at $0$ and $1$, choose $a\gt1$ and $b\gt1$.
