A supreme court puzzle

The first thing to realize about this problem is that we can consider each of the nine supreme court separately. If $X_i$ is the vacancy of the $i^\text{th}$ seat ($1$ if it’s vacant, $0$ if it’s occupied), then the quantity we’re after is:
\[
\mathbb{E}\left[ \sum_{i=1}^9 X_i \right] = \sum_{i=1}^9 \mathbb{E}\left[X_i\right] = 9\, \mathbb{E}\left[ X_1 \right]
\]and this follows by linearity of expectation. So it suffices to consider the problem of a one-person bench and multiply our final result by $9$.

General solution strategy

Let’s say that the government is “aligned” if the same party holds the presidency and the senate. Imagine the distribution of $X_1$ (vacancy of the first seat) over time. When the seat becomes occupied, $X_1 = 0$, we must wait some number of years that is uniformly distributed between zero and $40$ years before it becomes vacant again. When the seat become vacant, $X_1 = 0$, two things can happen:

1. If the government is aligned when the seat becomes vacant, then it can be filled right away with no wait required. The seat will remain filled for an average duration of 20 years (the expected value of the uniform distribution on the interval $[0,40]$).
2. If the government is not aligned when the seat becomes vacant then the seat will remain vacant until the next election. Then, if the election aligns the government, the seat will fill. Otherwise, the wait continues.

Senate elections happen every two years while presidential elections happen every four years. Every two years, there is an independent probability of $1/2$ that the government becomes aligned. This occurs when the senate aligns itself with the presidency. It doesn’t matter whether the president changes or not since the outcome of the senate race is independent of the outcome of the presidential race. Given repeated independent events each with probability $p$, the expected number of events before we encounter our first success is $1/p$. This fact was demonstrated in the post about the Monsters’ Gems puzzle. In this case, $p=1/2$ so we can expect to wait $2$ elections before the government becomes aligned. Here is a tally of the possible cases when the seat becomes vacant:

1. With probability $1/2$, the government is aligned and the wait is zero.
2. With probability $1/2$, the government is not aligned. In this case, we must wait an average of $1$ year until the next election. Then, we must wait an average of $2$ more elections. This will only incur a wait of $2$ years on average because we must only wait the time between both elections. If the second election aligns the government, we can fill the seat right away! So the expected wait is $1+2=3$ years.

Combining these two facts, we are left with a net expected wait of $1.5$ years during which the seat is vacant.

On average, the empirical expected number of vacancies in the long run is:
\[
\frac{1.5}{20 + 1.5} = \frac{3}{43} \approx 0.069767
\]If we account for all nine seats, we multiply this number by nine:

$\displaystyle
(\text{Expected vacancies}) = \frac{27}{43} \approx 0.628
$

So the expected number of vacancies is less than one.

If we make the approximation that the variables $X_i$ are mutually independent, the expected number of vacancies is a Binomial distribution with parameters $n=9, p=\tfrac{3}{43}$. Using this fact, we can compute an approximate distribution of long-term vacancies:

We can see that the vacancies are $0$ about half the time, and are rarely greater than $2$.

Note: We made an approximation in the solution above. If a supreme court justice’s term expires while the same government is still in power (before any other elections take place), then the seat will always be filled immediately since the government is still aligned. This effect only matters for up to two years into the justice’s term. If we account for this special case, it will slightly increase the average duration of a term and therefore slightly decrease the expected rate of vacancy. Hector Pefo has a more detailed solution on his blog where he works out all the details.