smc

By symmetry, the probability of the red ball landing in a higher-numbered bin is the same as the probability of the green ball landing in a higher-numbered bin. Clearly, the probability of both landing in the same bin is ${\sum }_{k=1}^{\infty }{2}^{-k}\cdot 2^{-k}={\sum }_{k=1}^{\infty }{2}^{-2k}=\frac{1}{3}$ (by the geometric series sum formula). Therefore, since the other two probabilities have to both the same, they have to be $\frac{1-\frac{1}{3}}{2}=\boxed{\frac{1}{3}}$. Note: the formula is $\frac{a_1}{1-r}$ where $a_1$ is the first term and $r$ is the common ratio. Derivation of the geometric series sum formula: Let $S=S=a_1+a_{1}r+a_1{r}^2+a_1{r}^3+...$ and so on to infinity. Then $rS=a_{1}r+a_1{r}^2+a_1{r}^3+...$ and so on to infinity. Notice that the terms in the second expression are the same as all the terms in the first EXCEPT for $a_1$. Subtract $S-rS=a_1$, factor $S\left(1-r\right)=a_1$, and finally $S=\frac{a_1}{1-r}$. Note: The formula only works if $r<1$; otherwise, the series will diverge to infinity or negative infinity.