Don't mind me; I'm on a math bender - Macleans.ca

# Don’t mind me; I’m on a math bender

As the paid-up holder of a Mainstream Media club card, can I warn the sportswriters away from making too much of the statistical fluke of all eight first-round NHL playoff series starting out tied through two games? The warning will arrive too late for some, but others may yet be saved.

As a landmark of NHL parity, the large number of 1-1 results in 2010 is not going to prove very useful. Imagine that game outcomes are statistically independent of each other and that the better team has a p chance of winning each individual game in the home team’s rink. If that’s the case, then the chance of a given series standing level after two games is 2(p)(1-p).

The 1-1 tie is always, for realistic values of p, the most common outcome. In a world of perfect parity—all teams are equal, no home-ice advantage, p = 0.5—half the series will be tied 1-1 after two games. And because the chance of the better team going up 2-0 is counterbalanced by a decreased chance of the other team going up 2-0, the overall chance of a tied series doesn’t drop off very fast as you depart from the parity condition, p = 0.5. For p = 0.6, about 48% of the series are still tied 1-1 after two games. (The better team is ahead in 36%, or 0.6²; the worse team is up 2-0 in 16%, about 0.4².)

But you can see that having eight series tied 1-1 will be incredibly rare even in the world of perfect parity. The probability of that happening in a given year will be the total product of the chances of a 1-1 tie in each of the series. Given an average overall value of p, the odds of all eight series starting out equal works out to, at most, (2(p)(1-p))8—a pretty small number, demonstrating the great flukiness of the “eight ties” outcome. Even in the perfect-parity world the expected frequency works out to 1 time in every 28, or 256, years. In the real world, the right average figure for p is probably around .54, giving us an “eight ties” year about 1 time in 269. In a fairly extreme non-parity world where the 1-4 seeds had an average 60-40 edge—that is to say, p = 0.6—the “eight ties” outcome would happen once every 355 years.

In other words, using this fluke as any kind of sign, indicator, or test for parity is about like insisting on reading a book only by the light of Halley’s Comet. You’d better have a comfortable chair. And plenty of kids, so they and their progeny can continue the observations (over several millennia) after you die in it…