Let A and B each be random real numbers chosen from the uniform interval (0,1).

Call Z the tenths place digit of A^{B}.

Find the probability distribution of Z.

We want the tenths digit of y^x. First find the probability density function of x log(y) which is Gamma[0,-z] for z<0 (here Gamma is the incomplete gamma function). Now integrate this over the intervals (log((n-1)/10),log(n/10)) for n=1,2...,10. Using this result you can get the cumulative sum of this expression to obtain the distribution function (cumulative distribution function) which can be written either in terms of the incomplete gamma function or the logarithmic integral function li. The logarithmic integral function form of the cumulative density function is then:

li((n+1)/10) log(10/(n+1))+(n+1)/10, for n=0,1...,8

In the case of n-> 9 take the limit of this and you get the required number of 1.

Software packages like Mathematica can evaluate this to any degree of precision we might require. Here are the values to a few digits:

{0.0254198, 0.0629942, 0.110477, 0.168225, 0.237525, 0.320654,

0.421456, 0.546953, 0.712901, 1.}