I have a state which is a list of probabilities for next states as follows:
$\left[0, \frac {1}{2}, 0, 0, \frac{1}{2}\right]$
I realise that the second element here actually has a $\left(\frac 2 7\right)$ chance of looping back to this state. So I can remove it like so:
$\frac 1 2 -\left(\frac 1 2 \times \frac 2 7\right)$
Which gives me $\frac 5 {14}$, which is correct in this case, but I might still have the wrong method for others
I want the other element to equal $\frac 9 {14}$, which I have been able to do, but not in the general case.
Hope this is clearer.
I am taking quite the long shot here, but how about
$ p_j' = (1-p_1') \cdot \dfrac{p_j}{\displaystyle\sum_{k=2}^n p_k} $,
where $ p_1, \dots , p_n $ are the original probabilities and $ p_j' $ the new ones. It keeps the sum at 1 and the "extra" probability is divided somewhat analogously.