What are the Odds? Canada and the Stanley Cup Playoffs
Another year has come and gone, with no
prospect of a Canadian team winning the Stanley Cup. This will be the 22rd year in a row, without
a win. This year comes with a special
twist, as none of the Canadian teams even made the playoffs, so we know for
certain that there can’t be a Stanley Cup winner based in Canada this year.
So, what are the odds of all 7 Canadian teams
missing the playoffs in the same year.
One might intuitively think that it is not all that unlikely - after
all, 16 teams have to miss the playoffs, and that means there were actually 7
American teams that missed as well as the 7 Canadian teams. That’s not really such a big deal, you might
think.
Actually the odds against this occurring are
pretty low. You can calculate it at
about 2 chances in 1000, or 0.0017 or 0.17%, depending on how you like to
describe it. There are several ways to
calculate this:
·
1 - The first way, is the
classic “coloured balls in an urn” technique.
In this case, we frame the problem as “you have an urn with 30 balls
(the number of teams in the NHL). 23 of
those are blue (U.S.) and 7 are red (Canadian).
What is the probability of drawing 16 balls, with all of them being
blue?”. And as we might remember from
mathematics that probability is:
o (23/30)X(22/29)
X(21/28) X(20/27) X(19/26) X(18/25) X(17/24) X(16/23) X(15/22) X(14/21) X(13/20)
X(12/19) X(11/19) X(10/17) X(9/16) X(8/15) = 0.0017.
o The
basic idea here, is that on the first draw there are 23 blue balls (U.S.
teams), out of 30 in all. So, that gives
23/30. On the second draw, there are now
22 blue balls, out of 29 in all, left in the urn. We keep doing that for all 16 draws (the
number of playoff spots), decrementing both numerator and denominator each
time. Then, we multiply out all of those
probabilities to get the overall probability.
·
2 - The second technique, is
to calculate the number of ways there are to form the numerator (how may ways
can you create a list of 16 teams from 23 U.S. teams), then calculate how many
ways there are to form the denominator (how many ways can you create a list of
16 teams from 30 teams). Then divide one
into the other, and you have your probability, which is again .017. You can work this out from the formula for
combinations, which is N!/(N-K)!K!, where N is the total number of items in the
list and K is the number you want to choose.
Or, you can use the excel COMBIN function.
·
3 - Alternatively, you can
write a little Monte Carlo program or set up a Monte Carlo spreadsheet in
excel. In this, you would set up a list
of 16 random numbers, then check them to see if the first in the list was less
than 23/30 (that would indicate a U.S. team in the first playoff spot), the
second was less than 22/29 (that would be a U.S. team in the second playoff
spot), and so forth, until all 16 spots were filled. For a given list of 16 spots, check whether
zero Canadian teams were included. Then,
repeat that a few hundred or a few thousand times, and calculate how many times
you had zero Canadian teams in a given list of 16 spots. You should find a small number of occasions
where that happens, about 1 in every 500 sets (0.0017) of 16 playoff lists (or
simulated seasons). That’s what my excel
spreadsheet gave.
The other question is “what are the odds of no
Canadian team winning the cup for 22 straight years?” . That’s a pretty low probability event as
well, at about .0053. One way to
calculate that is via the binomial theorem.
You can look that up in a table (lots of them can be found on the
internet, or you can find one in a stats textbook), or use the excel function
BINOM.DIST. The calculation can be
described as “how many times would you get 0 successes in 22 trials, assuming
the probability of success in any given trial was about 7/32” (actually, I used
.212 instead, to reflect the changing percentage of Canadian teams in the
league over the 22 years).
You can also set up a Monte Carlo for this,
as a check. The procedure is similar to
before:
·
Make a list of 22 random
numbers.
·
Test them to see if any of
them are less than .212.
·
If so, then a Canadian team
won in that run of 22 seasons.
·
If not, you have uncovered a
simulated streak similar to the one that we are living through.
When I did this, I found 3 occasions in my 22
season streaks, out of 500 trials, where no Canadian team won the cup. That’s about .006, very similar to what the
binomial theorem predicted.
You can do other things, like simulate this
procedure with playing cards instead of computer generated random numbers. Remove the King and Queen of clubs from the
deck, then deal cards from a shuffled deck until you hit a club. Record how many cards you had to turn over to
do this. I did this, and found only one
club-free run in 250 trials (equivalent to no Canadian team winning a Stanley
Cup). That represents a 0.004
probability of this event occurring, very similar to what the Binomial theorem
predicted.
So, we have two awfully unlikely events
occurring. It begins to stretch
credulity to consider this merely bad luck.
Some other theories have been propounded:
·
It’s related to the Canadian
dollar. But that doesn’t work, as the
Canadian dollar has varied considerably during this time period, sometimes
being lower and sometimes higher than the U.S. dollar, as the accompanying
graph shows (the graph shows how many Canadian dollars it would take to buy a
U.S. dollar, from about 1970 to the present day). Indeed, during the 1980’s Edmonton Oilers
dynasty, the Canadian dollar was quite low, and during the 2000-2010 part of
the recent Stanley Cup drought it was quite high.
·
Another theory states that
hockey players can’t stand the pressure of playing in the Canadian cities,
where the sport is taken very seriously by the fans. Therefore, the players
choke, basically. That theory falls down
in two ways:
o First, professional athletes are not given to
choking, especially so many athletes over such a long period.
o Second,
they should have been choking in hockey-mad U.S. cities like Chicago, Boston,
Detroit and New York, too. They didn’t ,
as all of those teams have won cups during the Canadian drought.
So, the only theories left, are:
·
conspiracy (the league let
the U.S. teams win, to help the U.S. marketplaces),
·
economics (Canadian teams can make as much money
losing as they can winning, so they don’t manage business affairs with winning
as a significant priority),
·
or plain bad luck.
Truly a conundrum.
==============================================
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