Josh Blackman recently noted that Judge Stephen Reinhardt has the "uncanny ability to be on the right panels," and asked what the odds are that he could serve on the three recent panels, one regarding California's marriage amendment, another on striking jurors on the basis of sexual orientation, and a third on other Ninth Circuit marriage cases.
The odds are about 1 in 1000.
That's slightly deceptive--it's not unique to these three cases. It's simply because the odds of being on any three random panels are 1 in 1000 in the Ninth Circuit.
But here's now the math works.
There are 29 active judges in the Ninth Circuit. The odds of being on any given panel are
1/29 + 1/28 + 1/27, or 10.72% 1/29 + 1/28(28/29) + 1/27(27/29), or 10.34%.
But there are also 16 senior judges, and one of them may sit on a panel with two active judges. In those cases, the odds are
1/45 + 1/29 + 1/28, or 9.24% 1/29 + 1/28(28/29), or 0.69%.
So assuming the odds of selecting the first judge are completely random, there is a 29/45 chance that the first set of odds applies, and a 16/45 chance that the second set of odds applies for an active judge. That means we have
10.72 * (29/45) + 9.24 * (16/45), or 10.2% 10.32 * (29/45) + 6.89 * (16/45), or 9.10%, that an active judge will be selected for a given panel.
If we're looking at three panels, we take those odds and raise them to the third power. That leaves the odds of serving on any given three panels as
0.106% 0.075%, or something a little over 1 in 1000.
There are a number of caveats. Not all judges are "on" at the same time, and some pick different months to be available for panels, so the number at any given time is often fewer than 29. Senior judges have a lower caseload, and their odds of being picked may or may not be as certain a figure. Sometimes visiting judges, such as district court judges or senior judges from other circuits, may sit on the panel. Not all of the vacancies were filled on the Ninth Circuit in the last couple of years, and the number of senior judges fluctuated slightly in that period, too.
Yes, there's deeply limited data. But that's the high-level math behind any given Ninth Circuit panel selection.
Update: Roger Ford tweets, for just Ninth Circuit active judges, "It’s 3/29 or (1/29)+(1/28)*(28/29)+(1/27)(27/29)." That's more accurate--it factors in the odds as to whether a given judge has been selected for the first or second slots on the three-judge panel. And that's about 10.3%, slightly lower odds than my original math, but still comes out to just about 1 in 1000.
Update II: I've attempted to fix the math... but this is the peril when it's been a decade since my last math class. Thanks for all the patience with my rough efforts.