Why the Braves series may (or may not) matter so much

The Nats are getting ready to start a three-game series against the Eastern Division’s second-place Atlanta Braves. Suppose you’re trying to explain to a skeptical friend why this series matters so much.

You: This series matters because the two competitors are playing face-to-face. Every Nats victory is a Braves loss, and every win by the Braves is a Nats loss.

Skeptical friend (SF): Ok, I get it that there aren’t any games where the teams stay the same. But if the Nats win a game from the Braves, that’s not any different than Sunday’s games when the Nats won and the Braves lost.

You: But the Nats have a chance to sweep the series and move 8 games ahead. That would pretty much knock the Braves out of the running.

SF: Or, of course, the Braves could sweep, cutting the gap to 2 games and making the division a real race. But the same things could happen no matter who the Nats and Braves were playing.  In next weekend’s series, the Nats could sweep the Phillies and the Braves could be swept by the Giants, or vice versa. Why is it so critical that the Nats and Braves are actually playing each other?

You: Ok, let me try to explain this mathematically. The Nats and Braves are pretty evenly matched teams. So let’s say the odds of winning any single game is 50-50.

SF: The odds probably aren’t exactly 50-50, depending on who’s pitching, but I agree they’re pretty evenly matched, so 50-50 odds is ok as a rough approximation.

You: Now what are the odds that the Nats sweep? If the games are independent trials, the odds of the Nats sweeping are (1/2)×(1/2)×(1/2)=(1/8). The odds of the Braves sweeping are also (1/8), so the odds that either the Nats or Braves sweep are (1/8)+(1/8)=(1/4).  There’s a one-in-four chance that either the Nats sweep the Braves, knocking them out of contention, or the Braves sweep the Nats, making it a very tight race.

Now, suppose the Nats and Braves are each playing series against another team, also with a 50-50 chance of winning.

SF: 50-50 odds seems like an ok assumption for a series between the Braves and the Giants, but the Nats should have better odds than that against the Phillies, even at Citizens Bank Park. But, go ahead, let’s hear what happens with 50-50 odds.

You: The odds of the Nats sweeping a series against another team is 1/8. The odds of the Braves being swept is also 1/8. (I’m assuming both teams are playing 3-game series—yeah, I know the Braves are scheduled to play four against the Giants, but bear with me. I’m just giving a math example, not talking about the actual schedule.) In contrast to the scenario when the Nats and Braves play each other, when they each play other teams they are independent events, so the odds that both the Nats sweep the series and the Braves are swept is only (1/8)×(1/8)=(1/64). Similarly the odds that the Nats are swept and the Braves sweep is also 1/64. So the odds of a pennant-race changing scenario where the gap widens or narrows by three games is only 1/32, rather than 1/4.

SF: Oh, I get it. If either the Nats or Braves win 2 to 1, it’s not much different than if they were playing against any other team. But there’s a much better chance that one of the teams will sweep the other and really make a change in the standings. And if that happens, the series will have been really important—maybe even a turning point in the pennant race.

You: Yep, you got it.

SF: So I guess these games are probably worth watching.

You: You know I’m going to be watching them. Let’s go Nats!