Lesson 10

Let us construct the transition matrix for this process. The probability of drawing a ball of a given color is proportional to the number of balls of that color remaining in the bag. Therefore, if we order the states as red, green, and blue, the transition matrix is \[\mathbf{T}= \begin{bmatrix}1/8 & 2/8 & 2/8 \\ 3/8 & 2/8 & 3/8 \\ 4/8 & 4/8 & 3/8\end{bmatrix}\] Initially, a red ball is drawn, so the initial distribution \(\mathbf{r}= (1, 0, 0)^{T}\). Now, to find the probability distribution after \(2\) steps, we simply calculate \[\mathbf{T}^{2} \mathbf{r}= \begin{bmatrix}15/64 \\ 21/64 \\ 28/64\end{bmatrix}\] Since we are interested in the probability of drawing a green ball, the answer is \(21/64\).