Ehrenfest's urns (two of them) have been loaded with a total of 6 balls. If the first urn now contains 1 ball, what is the probability that it will contain 2 balls after the next transition?
My attempt on this question was drawing out the transition matrix associated. I know it is a 6x6 transition matrix. I also believe that you would acquire the result by looking at the first row second item? But I'm confused on what the matrix would be filled with.
Thanks.
Actually it is a $7\times7$ transition matrix for the number of balls in the first urn.
If the indices run from $0$ to $6$ then $p_{i,i-1}=\dfrac{i}{6}$ and $p_{i,i+1}=\dfrac{6-i}{6}$ with all the other values $0$.
You want $p_{1,2}$.