About the probability function $ \alpha \binom{x+q}{q}$ In a uniformly random arrangement of $m$ zeroes and $p$ ones, let $T$ be the number of trailing zeroes.
Then $$P(T=t) = \frac{1}{\binom{m+p}{p}} \binom{m+p-1-t}{p-1} \hskip{1cm} t=0,1 \cdots m$$
Or, alternatively, calling $q=p-1$ and $U=m-T$
$$P(U=u) = \frac{1}{\binom{m+q+1}{q+1}} \binom{u+q}{q} \hskip{1cm} u=0,1 \cdots m$$
That this is indeed a valid probability function can be verified with the 
Hockey-stick identity.
I didn't find this (quite simple) probability function  in the usual lists.
Has it a name? What about its mean and other properties?
 A: This is a slightly disguised negative hypergeometric distribution, with $r=1$.
The hypergeometric and negative hypergeometric distributions are about sampling without replacement from a fixed pool, which you have here; the hypergeometric counts how many successes in a fixed number of trial, while the negative counts how many successes until a given number of failures.
Rather then "number of trailing zeros", instead take T as "number of leading zeros".  Then it's counting until the first "1" sampled spoils things.
The general PMF is
$$ \operatorname{PMF}(t) = \binom{t+r-1}{t} \binom{N - r - t}{K-t} \Big/ \binom{N}{K},$$ with $N$ the total, and $K$ the successes.
Substitute in the following:
$$
\begin{eqnarray*}
N &=& m + p \\
K &=& m \\
r &= 1 \\
\end{eqnarray*}
$$
And we have your PMF:
$$
\begin{eqnarray*}
\operatorname{PMF}(t) &=& \binom{t+1-1}{t} \binom{m + p - 1 - t}{m-t} \Big/ \binom{m+p}{m} \\
    &=& \binom{m + p - 1 - t}{p-1} \Big/ \binom{m+p}{m}
\end{eqnarray*}
$$
The mean is
$$\operatorname{Exp}(t) = r\frac{K}{N - K + 1} = \frac{m}{p+1}$$
and the variance is
$$
\begin{eqnarray*}
\operatorname{Var}(t) &=& r\frac{(N+1)K}{(N-K+1)(N-K+2)}\left[1 - \frac{r}{N-K+1}\right] \\
    &=& \frac{(m+p+1)m}{(p+1)(p+2)}\left[1 - \frac{1}{p+1}\right] \\
    &=& \frac{(m+p+1)m}{(p+1)(p+2)}\left[\frac{p}{p+1}\right]
\end{eqnarray*}
$$
