I suspect that the issue is that of "round off error" because of the large numbers involved (such as 145^6 = 9294114390625) when an attempt (internally) is made to match a uniform random number to an approximation of the cumulative distribution function. (Maybe scaling things using Horner's scheme might be helpful to reduce the round-off error.)
One approximation to get around this is to generate a list of values for the cumulative distribution function (with the associated values of DA) and then use interpolation. One can make the approximation more accurate but at the expense of more time creating the interpolation table. Here's an example:
cumulative = Table[{NIntegrate[Prob[x], {x, 110, t}], t}, {t, 110, 145, 1/100}];
rSample = Interpolation[cumulative];
Histogram[rSample[#] & /@ RandomReal[{0, 1}, 1000]]
