TITLE: A Query regarding the Halting Problem (Omega): Halting Probability for Given Input Size
QUESTION [6 upvotes]: I was studying the Halting Problem in context of the Probability and had a few doubts regarding it. Hope someone could help me out.
I am aware of the probability of a Random program halting on a Universal Turing Machine (with a randomly chosen input) is given by Chatin's Constant (which is normal and transcendental) number thus cannot be exactly computed.
My Query is the following:
Suppose one is given a Specific Universal Turing Machine Model (T) for doing all the computations. Also the following facts are given:

The size of the randomly chosen Program (P) is 'at most' (Pk) bits. 
The size of the randomly chosen Input (I) for the Program (P) is 'at most' (Ik) bits.

**The Program and Input are nothing but two randomly chosen strings.
Now given a random instance of a program and the Input for the Turing Machine (T) [with the size limits for Pk and Ik known]:
a. Is it possible for one to calculate the probability of any random  Pair halting on the Turing machine (T) ?
b. If not possible to calculate the exact probability (the value being transcendental) then the first few Most Significant Bits, let us say first Min(Pk, Ik) bits of the Probability i.e. just an aproximation ?
c. Is is possible to comment on a rather modified Probability Statement (and find an aproximation) which is as follows:
Probability(P, I, Lk) : Its defined as the Probability that a random  Pair will halt after exactly Lk Steps/Operations on the Turing Machine T ?
d. How do we go about finding it as the Chatin's formula is for an infinite Series?
Hope someone could help me out here.

REPLY [2 votes]: "
Hi Tarandeep, 
The halting probability for a random n-bit program certainly won't be transcendental -- in fact it's rational ( # halting progs / 2^n )! On the other hand, that probability clearly isn't computable given n as input, since if it was then you could use it to solve the halting problem. 
Hope that helps, 
Scott
"
Thanks to Prof. Scott Aaronson.