One bag contains 3 white &2 black balls, and another contains 2 white & 3black balls. A ball is drawn from the second bag & placed in the first, then a ball is drawn from the first bag & placed in the second. When the pair of the operations is repeated, the probability that the first bag will contain 5 white balls is?
I encountered this problem while practising probability, I took one case 2W1-1B2-2W1-1B2. And for this I got the probability to be 1/225 by multiplying the probabilities of the successive events and that was the answer too.
but can't there be more rather infinite cases of doing this like 2B1-1B2-2W1-1B2-2W1-1B2, in a similar way we can generate infinite cases.
Abbreviation used : 1B2 means taking a black ball from bag 1 to bag 2.
Kindly tell what I am thinking wrong about the rest of the cases,
Any help will be appreciated.
Best Answer
This is a short enough question that you could work through the two iterations on paper if you wanted to get a feel for the interactions between the bags of balls. I suggest drawing pictures and writing out each probability and the probability of the intermediate states. As an overview what should be clear is that for the first bag to contain five white balls after two iterations, both of the white balls have to be selected from the first bag and no black ones, and no white balls can be returned from the first one. So the case you described, 2W1-1B2-2W1-1B2, is the only one that will result in five white balls in the first bag and five black balls in the second, any other pattern of draws will result in a mixture of white and black balls in each bag.