Measuring Luck
In the story “A Dark Horse”, in the collection Northern
Gothic Stories, Daniel Foster, a gambler, has a difficult time with lucky
streaks, bad and good. He worries that
perhaps something more than the vagaries of random chance are at work, perhaps
even something diabolical.
Anyway, a
while back, I was locked into the deepest losing streak I'd ever known, maybe
the deepest losing streak anyone has ever known. At least that's how it seemed
to me.
I'd been six straight weeks
without a winning day. Hell, at one point I'd gone fiftyseven straight races
without seeing the cashier's window. The
odds against that must be astronomical.
A dead man could do better. I mean, pure dumb luck ought to count for
something. At any rate, I was feeling pretty desperate.
…
My luck changed dramatically
and all for the better. It's been the
hottest winning streak I've ever had, for all I know it's been the hottest
winning streak of all time.
I don't know and at this
point I really don't care. You see, this streak has been too much, too unreal
for me to feel comfortable with. I like
to win. Every gambler likes to win. Hell, everyone likes to win, gambler or not.
But this thing  I don't know. I think I preferred the losing streak.
The thing that's really got
me are those damned dreams. Every night it's the same thing. They follow the
script, the one I described before. I
get up and go to the desk. The dark man
refuses to tell me his name. We make a bargain, and we seal it with a drink. As
I turn to leave, I hear the name of a horse.
The next day that horse is on the card.
If I bet the horse, he wins. If I don't bet he loses. I've made over $200,000 in the last month
alone, and I'm not even trying. A horse
player's dream, right? I'd give it all up for a good night's sleep.
I think we have all been through something like this in our
lives, whether or not it involved gambling.
It can be exhilarating to be on a hot streak, and devastating to be on a
cold streak. It feels like the universe
has singled you out, for better or worse.
So, just how do you measure how likely or unlikely a hot streak or cold
streak is?
First off, it helps to have a well defined measure of success
or failure. In Daniel Foster’s case, it
was success at the track, which can be measured in percentage of wins and money
won or lost. In other cases, it can be
more nebulous  how do you measure “lucky in love”, for example?
So let’s stick with something easy. In this case, we will look at the 20 year
streak of ineptitude for Canadian NHL teams, in which they have not won a
single Stanley Cup (to be fair, it is really 19, after excluding the lockout
year). Is this just a streak of bad luck,
or is something else at work?
There are a number of ways to tackle this problem, so we
will go in order, from “common sense” methods, to physical modelling (via
playing cards), to computer modelling (via excel), to theoretical mathematical methods (via the binomial
theorem). Pick the one you understand
the best and like the best.
We begin by looking at what percentage of NHL teams were Canadian
during this time span, which turns out to be 21.2% overall, varying from a high
of 23.3% to a low of 20%. So, naively,
we would expect a Canadian team to win the Stanley Cup every 4 or 5 years,
which corresponds to the proportion of teams in the league. So, 19 years does seem like a pretty long
time to go without a Stanley Cup. Our
naïve statistical sense tells us this is about 4 or 5 times longer than we
would expect. Carrying on further
with our naïve statistical instincts, we might say:
This reasoning isn’t actually valid, but I think it gives a feel for how people reason about things like this. Plus, it does give an answer that is accurate, to a first approximation. It tells us that we wouldn’t expect a run like this very often.
·
There is roughly a 50% chance of a run of 5.
·
So, there is roughly a 25% chance of a run of 10
(half of 50%).
·
Then, there is roughly a 12.5% chance of a run
of 15 (half of 25%).
·
Giving a 6.25% chance of a run of 20 (half of
12.5%), more or less.
So, using our naïve probabilistic reasoning, we
think a run of 19 or 20 years without a Canadian team winning the Stanley Cup
is a pretty low likelihood event (at about 6.25%), but not alarmingly so. This reasoning isn’t actually valid, but I think it gives a feel for how people reason about things like this. Plus, it does give an answer that is accurate, to a first approximation. It tells us that we wouldn’t expect a run like this very often.
What’s our next effort to figure this out?
This time, let’s try an experiment, using a realworld
situation to model our problem. To do
so, I took a deck of playing cards, and let the suit Clubs represent Canadian
hockey teams (Montreal’s team is called the Canadiens’ Hockey Club, so I
thought it appropriate to let Clubs represent the Canadian hockey clubs). There are 13 Clubs in a deck of 52 cards, so
they represent 25% of the deck. If we
remove the King and Queen of Clubs, then that suit has 11 cards out of the 50
remaining, representing 22% of the cards in the reduced deck. That’s as close as we can get to the 21.2% of
Canadian teams in the NHL during the period in question, so we will go with that.
So, now we take our modified deck of playing cards, and
simulate the hockey problem by:
·
Shuffling the deck thoroughly.
·
Dealing cards out until we come to a Club,
counting the number of cards as we do so.
·
Recording the length of the run of nonClubs.
·
Repeat this as many times as you like (I did 100
trials).
My results are given below:
Run
Length

Frequency

Percent

1

20

20%

2

18

18%

3

6

6%

4

13

13%

5

10

10%

6

5

5%

7

8

8%

8

3

3%

9

4

4%

10

3

3%

11

3

3%

12

0

0%

13

3

3%

14

1

1%

15

1

1%

16

1

1%

17

0

0%

18

1

1%

19

0

0%

20

0

0%

20+

0

0%

Total

100

As you can see, the longest run was 18, not quite as long as
the number of years that Canadian teams have gone without winning a Stanley
Cup. So, it would appear that a run this
long is less likely than our naïve statistical sense told us. In fact, it appears that a run of 19 or 20
has a likelihood of coming up about once every 100 trials, at best.
For the heck of it, I tried this again, though this time I
didn’t reshuffle after hitting a Club, but dealt the deck on to
exhaustion. Basically, this was faster,
though it wasn’t as statistically rigorous, since in any one shuffled deck, the
short runs and long runs would be anticorrelated (sort of like the idea behind card counting
in blackjack). Anyway, here are those
results, this time using 250 trials:
Run Length Frequency Percent
1

47

19%


2

56

22%


3

27

11%


4

28

11%


5

25

10%


6

14

6%


7

13

5%


8

7

3%


9

6

2%


10

6

2%


11

3

1%


12

3

1%


13

5

2%


14

2

1%


15

1

0%


16

1

0%


17

2

1%


18

2

1%


19

1

0%


20

0

0%


20+

1

0%


Total

250


This time we hit one run of 19 and one run of 20+, so that’s
2 out of 250, or a little under 1%. Surprisingly
enough, the longest run was a run of 29.
It’s always weird to witness a such a low probability event happen
before your eyes, even if it means very little in the real world.
So, I think we can safely say from the experimental
evidence, that a run of 19 years without a Canadian team winning the Stanley
Cup is a pretty unusual event (odds are
on the order of 1%), if it is just a matter of random chance.
I also set up a Monte Carlo simulation in Excel. In that one, I simulated a 20,000 year NHL
history (yes, it is a bit excessive), and counted how many times a run of 19 or
more came, from random chance, given a Canadian team representation of 21.2% of
all teams. In 100 trials of this simulation,
the median percentage of runs of that length was a bit under 1.0%. That conforms nicely to our playing card
experiment.
Finally, we can look at this as a simple binomial
distribution probability problem (you may remember this from high school or
university math courses), with p=.21 and n=19, and consult a table like this
one:
Doing that, we also find the probability of a 19 year run of
no Canadian Stanley Cups to be about 1%.
The nice thing about the playing card simulation, is that it
is easy to understand and anyone can do it if they choose to spend half an hour
or so shuffling and dealing cards (perhaps while watching their favorite
Canadian hockey team scrub out of the playoffs). You don’t need a strong math background, just
common sense.
So, have we discovered whether the lack of success of
Canadian hockey teams is just one of those things, or is there something deeper
at work? I suppose it is still a
judgement call, but it does make you wonder.
Every year this goes on, makes you wonder even more. We’ll leave it at that for now.
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