Tuesday, February 19, 2008

Tonight's NHL hockey game between the New York Rangers and Montreal Canadiens featured a streak and a "counterstreak." The Rangers took a 5-0 lead, only to have the Canadiens score five straight goals of their own, producing a tie at the end of regulation. After a scoreless overtime period, the game went to a shootout, where Montreal prevailed.

Given that 10 goals total were scored in regulation, how likely is it that they would be distributed as five straight by one team followed by five straight from the other? Think of 10 boxes laid out side-by-side, numbered 1 through 10, and five pucks with Montreal Canadien logos (representing the 5 Montreal goals). In the actual game, the Montreal goals were the sixth, seventh, eighth, ninth, and tenth goals scored in regulation, thus filling boxes 6, 7, 8, 9, and 10 in our example.

The question then becomes, In how many possible different ways could the five Canadien pucks have been distributed into the 10 boxes? For example, Montreal could have scored the game's second, third, fourth, eighth, and ninth goals; or the first, fifth, seventh, eighth, and tenth. Clearly, there are many, many ways to distribute five objects into 10 boxes.

The mathematical tool we need to use is the "n choose k" principle, in this specific case, 10 choose 5. An online calculator is available for this purpose. The answer is that there are 252 possible different ways to distribute Montreal's five goals into 10 different positions. The actual sequence that occurred, as well as the hypothetical opposite one of the Canadiens scoring the first five goals of the game, would probably be of greatest interest, so the probability would be 2/252 or 1/126.

I did a similar analysis of the 2004 Boston Red Sox' comeback from down 0-3 in the American League Championship Series against the New York Yankees, and I received some criticism for the post hoc nature of the analysis. Applying the critic's reasoning to the present example, it doesn't seem very likely that any fans entered the arena wondering, "What's the probability that the Rangers will score the game's first five goals tonight, only to have the Canadiens answer with five straight?" Accordingly, many statisticians would probably take me to task for undertaking the analysis I performed. Fair enough, but my intent in conducting such analyses is simply to illustrate how various statistical formulas can be used in thinking about sports oddities.

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