An unexpected result involving sums of random numbers

[KeithB]

In Matlab with the following script:
sumcount = 0;
for i = 1:100000
s = 0;
count = 0;
while s <= 1
s = s+random('Uniform', 0,1);
count = count + 1;
end
sumcount = sumcount + count;
end

avgcount = sumcount/100000;

I get:
2.7195
2.7196
2.7190
2.7175
2.7183

Or, much better than yours. Of course this does not really "converge" to e.