Urn 1, Urn 2, …, Urn 5 each contain p white and q black balls. One randomly chosen ball is transferred from Urn 1 to Urn 2, next one randomly chosen ball is transferred from Urn 2 to Urn 3, and so on till finally one randomly chosen ball is transferred from Urn 4 to Urn 5. If the ball transferred from Urn 1 to Urn 2 is white, what is the probability the ball transferred from Urn 4 to Urn 5 is white?
I have arrived at a complicated expression for the aforementioned probability and am unsure regarding the methods I used to arrive at the result and the result itself and hence, any answers would be highly appreciated.
Method used: Since, it's given that the ball transferred from urn 1 to 2 is a white one, we simply have to consider all the different ways through which we can land a white ball into the fifth urn from the fourth. One way is that we get a white ball from urn 2 to urn 3 and another white ball from urn 3 to urn 4 and finally a white ball from urn 4 to urn 5 which the question demands. The probability of such an event would be ((p+1)/(p+q+1))^3. There would be three of more such possibilities wherein the transfers from 2 to 3 and the transfer from 3 to 4 would be white and black, black and white and ultimately, black and black, respectively, all ending at a white ball transferred from urn 4 to 5. We find the probabilities for the remaining three possibilities the same way as we did for the first one and sum all four of them to get our answer.