Probability of Dependent Events

$$\text{Justice Is (Color)blind}$$

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In the country of Fictopia, there are two different parties, Red and Blue. Their Supreme Court is supposed to have $11$ judges, but right now it has only $2$: one Red and one Blue. For each of the next $9$ months, a new judge will be added, with her party randomly matching either Red or Blue, with probability proportional to the relative makeup of the Court so far. (For example, in the first month, the new judge is equally likely to be Red or Blue, but in the second month, the probability will be $\frac{2}{3}$ in favor of the majority party represented on the court, and so on.)