How surprising is it that LeBron averaged 27 points, 7 rebounds, and 7 assists in his career but never had one game where he had 27 points, 7 rebounds, and 7 assists?

Joshua Brooks writes:

Listening to a Bill Simmons podcast and the stat came up (not confirmed) that over sixteen years Lebron averaged 27 points, 7 rebounds, 7 assists but never had one game where he had 27 points, 7 rebounds, and 7 assists. Bill and his guest thought that’s a crazy improbable stat and I guess maybe intuitively it does, but while I’m no statistician I’d imagine the three dimensional nature of the stat would mean it’s not all that in improbable. But what’s interesting is the category: what is the category of probabilities that seem intuitively (to the average Joe) as improbable but mathematically, are probable? And what are the characteristics of stats that fall into that category?

Hmmm . . . just to do a very crude calculation, suppose that in any game LeBron could score between 17 and 37 points, between 2 and 12 rebounds, and between 2 and 12 assists, with uniform distributions, then his probability of scoring exactly 27/7/7 would be approximately 1/2000. It turns out that he’s played approximately 2000 games, so it wouldn’t be a surprise if he’s ever had a 27/7/7 game but it’s not a surprise if it’s never happened. (I don’t know if it’s really never happened; I’m just relying on a third-hand report here.) I guess we should really bump up the probability a bit here, given that 27, 7, and 7 are in the middle of the distribution, but even if an appropriate probability is 1/1000, it’s still no shocker for it not to happen in 2000 games.

Brooks points to this clip where Simmons describes it as a “crazy” stat, but I agree that it’s not so crazy as all that. Bill Simmons is entertaining and thought provoking, but he’d never be mistaken for a quant.