All American NHL Playoff Odds

Let me preface this by saying that I'm not a hockey fan. Those of you who know me might be confused right about now as to why I'm writing a post about the NHL... let me warn you right now, this post has very little to do with hockey and everything to do with math.

My dad sent me a text earlier today - he had tweeted about the odds of every team in the NHL Stanley Cup Playoffs being American and wanted me to check his math.

"odds against this happening by chance are 593 to 1 ... look at standings... warm climate teams dominate in new nhl" - @JimDonovan1962

The short answer:
He's wrong. The odds are ²⁄_1,715, or ¹⁄_857.5

The long answer:
What my dad did was simply calculate the likelihood that 16 out of the 30 teams were American. He did this by calculating the number of possible ways that you can make 16 teams out of the 23 American teams, and divided that by the number of ways that you can make 16 teams out of the 30 total teams. (23 choose 16 divided by 30 choose 16)

The error in his judgement was the way that teams qualify for the playoffs. After the recent NHL changes, there are four divisions split between two conferences. The top three teams in each division make the playoffs, and of the remaining teams in each division, the top two are given wild card spots. After you take this into account, the math, albeit longer, actually becomes easier to follow.

In the Eastern Conference:
Metropolitan Division: The Metropolitan Division is relatively easy to calculate. All eight teams in this division are American, which means we never need to worry about one of the three playoff spots going to a Canadian team.

Atlantic Division: There are eight teams in the Atlantic Division - three Canadian and five American. That means that there is a ⁵⁄₈ chance that the first playoff spot goes to an American team, a ⁴⁄₇ chance for the second, and a ³⁄₆ chance for the third. Multiplying those together, we find that the total probability that all three of the playoff spots go to American teams is ⁵⁄₂₈.

Wild Card Spots: Out of the 16 total teams in the Eastern Conference, six already have playoff spots. That leaves ten teams (three Canadian and seven American) competing for the two wild cards. The probability of the first wild card going to an American team is ⁷⁄₁₀, and the probability for the second is ⁶⁄₉. Multiplying together, we get that the probability is ⁷⁄₁₅. 

Total for Eastern Conference: Multiplying together the probabilities above, we get ¹⁄₁ × ⁵⁄₂₈ × ⁷⁄₁₅ or ⁷⁄₈₄.

In the Western Conference:
Central Division: The Central Division has seven teams, six American and one Canadian. That means that the probability of all three spots going to American teams is: ⁶⁄₇ × ⁵⁄₆ × ⁴⁄₅, or after multiplying those together: ⁴⁄₇.

Pacific Division: The Pacific Division is made up of three Canadian teams and four American teams. Doing the same calculations as above, we get: ⁴⁄₇ × ³⁄₆ × ²⁄₅ or ⁴⁄₃₅

Wild Card Spots: The Western Division only has 14 teams, and again, six already have playoff spots. Of the remaining eight teams, four are Canadian and four are American. The probability of the first wild card going to an American team is ⁴⁄₈, and the probability for the second is ³⁄₇. Multiplying together, we get ³⁄₁₄. 

Total for Western Conference: Just as with the Eastern Conference, we multiply the probabilities from each section above to get ⁴⁄₇ × ⁴⁄₃₅ × ³⁄₁₄ = ²⁴⁄_1,715

Total for entire NHL: Here we simply multiply the probabilities from both the Eastern and Western Conferences to get ⁷⁄₈₄ × ²⁴⁄_1,715 = ²⁄_1,715. That means that the odds of every team in the NHL Playoffs being American are 2 in 1715, or roughly 1 in 857. 

Thanks for reading.