  We analyze the left-tail asymptotics of deformed Tracy-Widom distribution
functions describing the fluctuations of the largest eigenvalue in invariant
random matrix ensembles after removing each soft edge eigenvalue independently
with probability $1-\gamma\in[0,1]$. As $\gamma$ varies, a transition from
Tracy-Widom statistics ($\gamma=1$) to classical Weibull statistics
($\gamma=0$) was observed in the physics literature by Bohigas, de Carvalho,
and Pato \cite{BohigasCP:2009}. We provide a description of this transition by
rigorously computing the leading-order left-tail asymptotics of the thinned
GOE, GUE and GSE Tracy-Widom distributions. In this paper, we obtain the
asymptotic behavior in the non-oscillatory region with $\gamma\in[0,1)$ fixed
(for the GOE, GUE, and GSE distributions) and $\gamma\uparrow 1$ at a
controlled rate (for the GUE distribution). This is the first step in an
ongoing program to completely describe the transition between Tracy-Widom and
Weibull statistics. As a corollary to our results, we obtain a new
total-integral formula involving the Ablowitz-Segur solution to the second
Painlev\'e equation.
