CS2A - April 2023 - Question 1

[Michael Ardill]

Is anybody able to provide a high level explanation of the solution to April 23 Qn 1ii / iii please? Even with the examiners report I just can't follow the logic / get my head around it. Thanks!

 

[Andrew Martin]

Hi Michael

I'd approach this by trying to think about the different possible transitions.

Say we are currently in state i. This means there are i white balls in box A. As there are n balls in each box and n total white and black balls, this means there are n-i black balls in box A, i black balls in box B and n-i white balls in box B.

If we select a black ball from box A and a while ball from box B, then this means the process goes from state i to state i + 1. The probability of selecting a black ball from box A is (n-i) / n. The probability of selecting a white ball from box B is (n-i) / n. So, the probability of this happening is (n-i)^2 / n^2.

Note that the above holds as long as i = 0, 1, ... n-1. The probability formula still makes sense for i = n (in that it gives 0) but I'd be careful here. There is no state n+1 and so the probability of going from state n to state n + 1 is really undefined rather than being 0.

See if this helps get you started on the other possible transitions also.

Andy

 

[Michael Ardill]

Hi Andy

Thanks - that's really helpful!