Probability of drawing a white ball after successive transfers

[Kakashi]

Homework Statement

Each of k jars contains m white and n black balls. A ball is randomly chosen from jar 1 and transferred to jar 2 then a ball is randomly chosen from jar 2 and transferred to jar 3,etc. Finally a ball is randomly chosen from jar k. Show that the probability that the last ball is white is the same as the probability that the firs ball is white.

Relevant Equations

Total probability law

$$ P(\text{The event that the randomly chosen ball from jar 1 is white which is transferred to jar 2})=\frac{m}{m+n} $$
For the last transfer there are two disjoint events:

Conditioned on the event that the ball transferred from jar k-1 to jar k is white the probability that the randomly chosen ball from jar k is white is $$ P(W_{k}|W_{k-1})= \frac{m+1}{m+n+1} $$.

Conditioned on the event that the ball transferred from jar k-1 to jar k is black the probability that the randomly chosen ball from jar k is white is $$ P(W_{k}|W_{k-1}^{c})=\frac{m}{m+n+1} $$.

$$ P(W_{k})=P(W_{k}|W_{k-1})P(W_{k-1})+P(W_{k}|W_{k-1}^{c})P(W_{k-1}^{c})=\frac{m+1}{m+n+1}P(W_{k-1})+\frac{m}{m+n+1}(1-P(W_{k-1}))=\frac{m}{m+n+1}+\frac{P(W_{k-1})}{m+n+1} $$

How can I determine $$ P(W_{k-1}) $$?

 

[PeroK]

Can you think of another way to tackle the problem? Using a common mathematical idea.