<s>[INST] Probability with two boxes and balls There are $10$ white and $10$ black balls in the first box, and $8$ white and $10$ black balls in the second box. Two balls are put from the first to the second box, then we choose one ball from the second box. What is the probability that the chosen ball from the second box is white?
From the first box, we can choose two balls in ${20\choose 2}=190$ ways. Possible combinations are $\{BB,BW,WB,WW\}$.
After putting two balls from the first to the second box, there are total $18$ balls in the first, and $20$ in the second.
From the second box, we can choose one ball in ${20\choose 1}=20$ ways.
How can we find the probability that the chosen ball from the second box is white? 
 [/INST] Case $1$: $2$ white balls are put from first box to second and one ball chosen from the second box is white

probability $=\dfrac{\dbinom{10}{2}}{\dbinom{20}{2}} \times
 \dfrac{10}{20}$

Case $2$: $2$ black balls are put from first box to second and one ball chosen from the second box is white

probability $=\dfrac{\dbinom{10}{2}}{\dbinom{20}{2}} \times
 \dfrac{8}{20}$

Case $3$: $1$ white ball and $1$ black balls are put from first box to second and one ball chosen from the second box is white

probability $=\dfrac{\left[\dfrac{\dbinom{10}{1}\dbinom{10}{1}}{2!}\right]}{\dbinom{20}{2}} \times  \dfrac{9}{20}$

Sum of these three cases will give required probability
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