The probability of a randomly generated floating point number is in a certain range

Question

Let's say I randomly generate a number and check if it is in a certain range. For integer, it's simple. For example with unsigned 8 bits, the randomly generated number can be in range (0 - 5 inclusive) with the probability of (6/2^8).

My question is how can I calculate the same thing with floating point number. For example, when I just randomly generate 32bits, what is the probability that the number is within a range of -10.0 and 10.0?

Answer 1

Assuming a binary representation, the probability can be computed for ranges [2^n,2^n+1)
For example, if exponent is on 11 bits, probability is 1/2^12 (taking sign into account)

Inside such interval, it's possible to consider a uniform density of floating point.

Then I guess you could try and decompose your interval into such powers of 2 boundaries.
Then compute the probability of each interval, and sum them all...
Assuming there is a IEEE-754-like denormal representation, for the smallest possible exponent e, interval is [0,2^e[
So this should give you a rather simple procedure, but I see no simple formula.

For very accurate probabilities, you'll have to deal with the significand bit pattern of nearest representable float inside the interval for lower and upper bounds.