# Probabilistic Turing Machine Transition Relation vs Transition Function

In wiki, PTM are defined with two transition functions and a fair coin toss which determines which one is used at each step. One can also define them, much like NDTM, using a transition relation where in each step the machine chooses from its next possible configurations using a fair probabilistic choice. My question is: how are these equivalent? If there is, say 10 possible next configurations at a given step how could you encode this in a PTM which only has 2 transition functions?

It is impossible to generate a random number uniformly between 1 and 10 using coin flips using worst-case finite time. To see this, note that if the worst-case running time is bounded by, say, $$k$$, then there are $$2^k$$ possible coin flips, so $$2^k$$ possible outcomes, which is never a multiple of $$10$$, so we can never assign numbers between $$1$$ and $$10$$ to the $$2^k$$ outcomes in a uniform (fair) manner.
2. Worst-case finite running time -- this definition is used in the complexity class BPP. In this case, the above translation is not good as it will not preserve worst-case running time. Here we need a different translation which instead tolerates some error: we cut off the procedure to generate a random number between 1 and 10 after a certain constant $$k$$ number of iterations, and we keep track of the error involved. Since $$k$$ can depend on the input, we can make it large enough that the error is small enough that we don't care, at least for the purposes of the complexity class BPP, which is defined to be problems that can be solved to within a small probability of error.