Probabilities for Jar 2:

Probability Jar 2 gets a black is n/(m + n)

Probability Jar 2 gets a white is m/(m + n).

If Jar 2 gets a black then it has m white and n+1 black. If Jar 2 gets a white then it has m+1 white and n black.

Probabilities for Jar 3:

If a black got put into Jar 2,then:

Probability Jar 3 gets a black is (n+1)/(m+n+1)

Probability Jar 3 gets a white is m/(m+n+1).

If a white got put into Jar 2, then:

Probability Jar 3 gets a black is n/(m+n+1)

Probability Jar 3 gets a white is (m+1)/(m+n+1).

So the probability of getting a white from Jar 3 is:

.

Similarly the probability of getting a black from Jar 3 is .

A tree diagram makes it all very clear.

Obviously exactly the same argment can now be used to get the probaility of getting a white or black black from Jar 4, 5, 6, ...... k.