Does the probability of occurrence of a number remain same in bit level

Question

Say, a number x occurs with probability p.

x's binary representation be ABCD.

So, does each of A,B,C or D is set with probability p?

Answer 1

No. Let the range be $[0,15]$, which can be represented in four bits. If we select $x$ randomly from the range, we have $x=7$ (or any other value) $ \frac 1{16}$ of the time. Each bit is set $\frac 12$ of the time among the $16$ numbers.

Answer 2

Binary is either 0 or 1, so 4 bits gives $2^4=16$ possible combinations. So probability is $1/16$. But i dont think that answers your question.

Unless I've misunderstand your question:

Your number may as well just be an event E, which you say has probability $q$. Regardless of how you represent that event, in base 2 or 10 or whatever, it is still the same event so will still have probability $q$.