June 8, 2010
Partially inspired by Mike's question and partially inspired by hearing one too many commentator offer an off-the-wall statistic, I wanted to see what the odds of a Blackhawk's victory really look like. But first, allow me to digress.
Plinko is one of my favorite games on The Price is Right. A contestant stands at the top of an inclined plane and tries to slide a round "Plinko chip" into one of nine slots at the bottom of the board that determine the cash prize awarded to the contestant (Fig. 1). The challenge is that the inclined plane is hexagonally patterned with pegs, which reroute the Plinko chip in directions that the contestant can not easily predict. In this context, it is useful to think of the unpredictable path of the Plinko chip as a sequence of coin flips. Flip a head and the Plinko chip moves to the right. Flip a tail and the Plinko chip moves to the left.
In its most abstract representation, you might think about the outcome of a playoff series, say the Stanley Cup, in exactly the same manner. During a particular seven-game playoff series, each team has some probability of winning each game, so you can trace the path through Playoff Plinko in much the same way that a Plinko chip falls through the pegs. Flip a head and the team with the better record wins. Flip a tail and the team with the better record loses.
Rather than looking at the path of one Playoff Plinko chip, I aggregated data over all playoff series since the 1987-88 season to look at a phenomenological view of Playoff Plinko (Fig. 2). When viewing the data in this manner, a few non-obvious patterns emerge that make Playoff Plinko even more interesting than its game show brethren.
- Playing at home is a huge advantage for the team with the better record at the end of the regular season (the "better team"). During Games 1, 2, 5, and 7, the better team wins over 60% of the time. This effect is so strong that the better team is sweeping the series after two games 36.2% of the time, or 11% larger than you would expect if this were a typical (coin flipping) game of Plinko.
- Playing at home for the worse team generally only mitigates the skill of the better team, but it does not give them an overwhelming advantage. This is visually apparent by noting that the probability of each team winning during Games 3 and 4 are generally close to 50%, or what you would expect if this were a typical (coin flipping) game of Plinko.
- There is a discernable "elimination effect" in Game 6 where the team that is leading 3-2 is at least 13% more likely to win Game 6. This is particularly interesting in light of #2 above, which would suggest that each team would be equally likely to win Game 6. Unfortunately, this means that it is considerably less likely that Mike will see Toews hoist Lord Stanley's cup, but hey, at least the Blackhawks would be Stanley Cup champions.
There are a few other interesting things to note from Playoff Plinko. For instance, if you 'Toggle' to see Playoff Plinko sized by absolute number, it is interesting to see how rare it is that the worse team sweeps the series. If you notice anything else or have any complaints, suggestions, or general comments for Playoff Plinko, please let us know!