Lets say I have a set of people A,B,C,D and they will each be picked but their order matters. The probability that A will be picked first is 70%. And C getting picked 1st is only 12%. According to the matrix below.
A B C D
1 0.70 0.13 0.12 0.05
2 0.15 0.65 0.13 0.07
3 0.10 0.15 0.60 0.15
4 0.05 0.07 0.15 0.73
How do i recalculate the matrix of probabilities if C is chosen first, for instance? A should be more likely to be picked second than B, given that C was chosen 1st.
You first have to determine whether or not the process is memoryless. If it is, meaning that regardless of what happened until time $t_n$, at time $t_{n+1}$, A has a 70% chance of being picked first, 15% chance of being picked second, D has a 73% chance of being picked last, etc, then this is an example of a Markov chain and the matrix you show is its transition matrix. If so, then it doesn't matter who got picked at time 1. At time 2, the transition matrix remains the same. If it isn't memoryless, I do not think you have enough information to solve the problem, since you are only given one matrix which transitions from time $n$ to time $n+1$.Edit Upon further review, I believe that I made a mistake. I was assuming that one can go from any state to any state. After rereading your question, it seems that all of A, B, C, and D needs to be selected in any iteration before the process can start again, which makes it most definitely not a Markov chain. In that case, I believe the proper way to calculate the next step (still assuming independence) is to use the normalized submatrix obtained by removing the already selected point.
For example, assume that C is already chosen. The submatrix that remains is: $$ \left[ \begin{array}{cccc} 0.70 & 0.13 & \Diamond & 0.05\\ 0.15 & 0.65 & \Diamond & 0.07\\ \Diamond & \Diamond & \Diamond & \Diamond\\ 0.05 & 0.07 & \Diamond & 0.73 \end{array} \right] $$ Which becomes $$ \left[ \begin{array}{ccc} 0.70 & 0.13 & 0.05\\ 0.15 & 0.65 & 0.07\\ 0.05 & 0.07 & 0.73 \end{array} \right] $$
And when normalized $$ \left[ \begin{array}{ccc} 0.795 & 0.148 & 0.057\\ 0.172 & 0.747 & 0.080\\ 0.059 & 0.082 & 0.859 \end{array} \right] $$
Giving a 79.5% chance for picking A second after picking C first.