suppose i have a fair coin and two urns in urn 1 there are 1 red and 9

Question

Suppose I have a fair coin, and two urns. In urn 1, there are 1 red and 9 blue balls; and in urn 2, there are 9 red and 1 blue balls, identical except for their colors. First, I toss the coin once. If the outcome is heads, I take urn 1, and if the outcome is tails, I take urn 2. I use the selected urn for the entire process.Every round n = {1,2,3,...}, I randomly pick a ball from the selected urn.Then I return the ball back to the urn for the next round. Random processX[n] is defined as the total number of red balls observed in the first n rounds,divided by n. For instance, suppose the coin turns up tails, which means I will use urn 2. Let'ssay in rounds 1 to 5, the balls I pick are B-R-R-B-R, in order. Then, X[1]=\frac{0}{1}, \quad X[2]=\frac{1}{2}, \quad X[3]=\frac{2}{3}, \quad X[4]=\frac{2}{4}, \quad X[5]=\frac{3}{5} . Find E[X[n]]. Find Rx (n1, 1₂). Is X[n] ergodic in the mean?