Q:

Probability of mutation at the same site

In two identical, but genetically isolated DNA sequences (which are essentially strings of so-called nucleotides - A, T, G, C), for example:
seq1    ATAGTCGATGCTAGCTAGTC
seq2    ATAGTCGATGCTAGCTAGTC

If those sequences are part of a larger genome, and the probability of mutation at any one base (for each of the sequences) is 1e-6, and seq1 mutates at the first position, e.g. A>G:
seq1    GTAGTCGATGCTAGCTAGTC
seq2    ATAGTCGATGCTAGCTAGTC

Then am I correct in saying that the probability of getting a mutation at the same position in seq2 is 1e-6*1e-6? 
If so, is the probability of getting a mutation at the same position with the same of the four nucleotides 1e-6^2*1/4? i.e.
seq1    GTAGTCGATGCTAGCTAGTC
seq2    GTAGTCGATGCTAGCTAGTC

Then for this to occur again at a second position for both sequences, e.g. T > A:
seq1    GAAGTCGATGCTAGCTAGTC
seq2    GAAGTCGATGCTAGCTAGTC

Is the probability ((1e-6^2)/4)^2? 
And for a third position ((1e-6^2)/4)^3? e.g. G > C
seq1    GAACTCGATGCTAGCTAGTC
seq2    GAACTCGATGCTAGCTAGTC

And for a fourth position ((1e-6^2)/4)^4? etc.
I may be flawed in my thinking, but if you could correct me it would be grately appreciated. 

A:

Then am I correct in saying that the probability of getting a mutation at the same position in seq2 is 1e-6*1e-6?

Not quite. Be careful about your phrasing, or rather, how you think about conditional probability. Here's what we can say:

The probability of a mutation at position 1 in seq1 is 1e-6.
The probability of a mutation at position 1 in seq2 is 1e-6 (not 1e-12).
The probability of a mutation at position 1 in both seq1 and seq2 is 1e-12 (that is, 1e-6 * 1e-6).

That last one arises because if two events are independent, the probability of them both happening is the product of their probabilities. This works for any number of events, so, for example, the probability of a mutation at positions 1, 2, 3, and 4 in both seq1 and seq2 is 1e-6 ^ 8.
Something to be aware of, if you aren't already, is that these probabilities are for a prespecified position. For example, the probability that seq1 and seq2 will share a mutation on any position is much larger (and increases with the size of the genome) than the probability that they'll share a mutation at position 1 in particular.

