[mitthrowaway2]

There are several other objections posted, but only one that really refutes the heart of your claim.

To judge what set of numbers you're more likely to end up with, you need to specify a probability distribution. Without any specific information to prefer one number over another, you want the highest entropy distribution. It turns out that the most "natural" probability distribution for x extending between 0 and infinity is not uniform over x, but over its logarithm; therefore, by symmetry arguments, the probability is actually more like 50% that you draw a number between 0 and 1.

And it turns out that floating point numbers more or less respect this property.