Suppose we have a transition probability matrix, P, given by ________.

[ eginbmatrix 0 rac23 rac13 rac12 0 rac12 rac12 rac12 0 endbmatrix ]

with three states, X_n = 0, 1, 2. Using the formula for n-step transition probability, which is ________.

[ Pⁿ_ij = sum_l=0ⁱnfty Pᵐ_il P^(n-m)_lj ]

where ( Pⁿ_ij ) denotes the n-step probability of transitioning from state i to state j, how can we calculate ( P³_00 ) and ( P²_23 )?