Event of pulling a white ball in first draw is not independent of pulling a 2nd white ball

Question

You have two urns. Urn One contains 6 white balls and 2 black balls. Urn Two contains 7 white balls and 8 black balls. You pick a random urn; you get Urn One with probability 2/3. You pull out, sequentially with return, two white balls.

Show that for Urn One, the event of pulling out a white ball in the first draw is not independent from the event of pulling out a white ball in the second draw without return.

How do I tackle this problem. This isn't a homework question, I found it in a paper and it caught my attention.

Answer 1

If I'm understanding you well, the experiment only involves taking two random balls from Urn One.

To check if two probabilities are independent, you can use any definition of independence - they are equivalent. In this case, would just compute probability of the second ball being white if the first one has been white, and compare it the the probability of the second one being white if the first one has been black. If both events are independent, both conditional probabilities would be the same - that is, probability of the second event doesn't depend on whether the first has taken place.

If the first ball has been white, when you are going to take the second one there are 5 white balls and 2 black balls in the urn (there were 6 white balls at the beginning, but you removed one when you took the first ball).

P(second white|first white)=5/7

But if the first ball were black, there are 6 white balls and one black ball in the urn.

P(second white|first black)=6/7

Please notice that the event "first black" would be the same as "not first white".

Then, since P(second white|first white)<>P(second white|not first white), the events "second white" and "first white" are not independent.