I don't know why I can't solve this problem using Polya's method .$$$$ There are two boxes,one of which contains 5 black and 7 white marbles and the other have 2 black and 2 white marbles .randomly choosing two marbles form the first mentioned box (without seeing the colors ) and putting them in the second box.now we choose two marbles out of the second box,find the probability that both of them are white. $$$$ probability that the both chosen marbles form the first box are white
p(2W1)=$ \frac{\binom 72}{\binom {12}2}$
probability that the both chosen marbles form the first box are black
p(2B1)=$ \frac{\binom 52}{\binom {12}2}$
probability that the chosen marbles form the first box are white and black
p(W1B1)=$ \frac{\binom 71\binom 51}{\binom {12}2}$
and by solving this : p(2W2)=p(2W2|2W1)p(2W1)+p(2W2|2B1)p(2B1)+p(2W2|W1B1)p(W1B1)
P(2W2)=$\frac{31}{165}$
I get different answer when solving the problem using Polya's method that's
$ \frac{\binom 22}{\binom42}$
This is wrong:
p(W1B1)=p(2W1)=$ \frac{\binom 71\binom 51}{\binom {12}2}$
The probability that you choose one black and one white marble (from the first box) is the sum of the the probability that you choose a black marble first and then a white one and the probability that you choose a white marble first and then a black one:
$$\frac 5 {12}\times \frac 7{11}+\frac 7{12}\times \frac5{11}.$$
You can calculate the rest.