Kakashi
- 31
- 1
- 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 $$ P(W_{k-1}) $$?