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 first ball is white , i.e. , it is m / ( m + n ) .
Could you help me?
Let $A_j$ be the event that the $j$th ball transferred is white.
Claim: $P(A_j)=m/(m+n)$ for $j=1,2,\ldots$.
Proof by induction on $j$. Clearly true for $j=1$. Now assume true for $j=k-1$. Then \begin{align*} P(A_k)&=P(A_{k-1})P(A_k\mid A_{k-1})+P(A_{k-1}^c)P(A_k\mid A_{k-1}^c)\\ &=\frac{m}{m+n}\,\frac{m+1}{m+n+1}+\frac{n}{m+n}\,\frac{m}{m+n+1}\\ &=\frac{m}{m+n}\bigg(\frac{m+1}{m+n+1}+\frac{n}{m+n+1}\bigg)\\ &=\frac{m}{m+n}. \end{align*}