Probability of blue eyes in Canada's PMs

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TL;DR
The last ten prime ministers of Canada have all had blue eyes.
Mark Carney (2025-): Blue
Justin Trudeau (2015-2025): Blue
Steven Harper (2006-2015): Blue
Paul Martin: (2003-2006): Blue
Jean Chretien (1993-2003): Blue
Kim Campbell (1993): Blue
Brian Mulroney (1984-1993): Blue
John Turner (1984): Blue
Joe Clarke (1979-1980): Blue
Pierre Trudeau: (1968-1984): Blue

Census Canada does not collect data on eyes colour, but schools do. According to this census, 26-28% of school age children have blue eyes.
That is a little higher uncited accounts of Canadian eye-colour, which is estimated to be between 20 and 25% blue.

25% is equivalent to rolling a 4 on a D4. Rolling ten fours consecutively is (1/4)^10, or 1 in 1,048,576.
20% is equivalent to rolling a 5 on a D5. Rotting ten fives consecutively is (1/5)^10, or 1 in 9,765,625.

My thunder has been struck.
My gob: smacked.
My flabber? Ghasted!

I should have posted this in the Conspiracies Forum.
 
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What you are seeing is P(blue eyes | elected as PM). That is not the same as P(blue eyes). You are showing that there is a big bias for blue eyes being elected.
I'm not surprised except at the magnitude of the bias. The next question is how the statistics of the bias can be evaluated.
 
FactChecker said:
The next question is how the statistics of the bias can be evaluated.
With a chi squared test. But you need to know how many non-blue eyed PMs there were also.
 
Dale said:
With a chi squared test. But you need to know how many non-blue eyed PMs there were also.
Examining the dates, it looks like they have been all blue since 1968. I would be skeptical of the assumption that tendencies before 1968 are still valid now. We might have to go back before color TV was widespread to get enough numbers, and that would clearly add a confounding factor.
 
Dale said:
... you need to know how many non-blue eyed PMs there were also.
That would be interesting to know too, but it does not take away from the odds as we know them.

I'm imagining a fair D20, labeled blue on four or perhaps five faces. The other faces are appropriately labeled with brown, green and mixed (eg. hazel or grey).

No matter how many times the other colours had come up in the previous 14 throws, it would still defy logic intuition that blue came up the last 10 times in a row.

Two possibilities:
  • it is a fabulous 1-in-a-million to 1-in-9-million coincidence
  • it is not a fair die i.e. not a random chance. That means there is some other factor influencing the die rolls - no matter what occurred previously. (And therefor, somewhow, eye-colour is correlated with being PM)
Regardless of what previous 14 throws were - whether they were all brown, or all blue or a mix - we are still faced with an event with at least 1-in-a-million to 1-in-9-million odds.
 
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DaveC426913 said:
TL;DR: The last ten prime ministers of Canada have all had blue eyes.


That is a little higher uncited accounts of Canadian eye-colour, which is estimated to be between 20 and 25% blue.

25% is equivalent to rolling a 4 on a D4. Rolling ten fours consecutively is (1/4)^10, or 1 in 1,048,576.
20% is equivalent to rolling a 5 on a D5. Rotting ten fives consecutively is (1/5)^10, or 1 in 9,765,625.
The only analysis I can add is that any Canadian politician without blue eyes should consider getting blue-tinted contacts. :-)
 
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FactChecker said:
The only analysis I can add is that any Canadian politician without blue eyes should consider getting blue-tinted contacts. :-)
True!

And of course, as we know, correlation is not causation. We cannot rule out the possibility that being elected PM is the cause - and eye colour is the effect!

Clearly, something about being PM (since 1968) turns a person's eyes blue!
 
DaveC426913 said:
That would be interesting to know too, but it does not take away from the odds as we know them.
It does actually. This is something that often trips people. Getting a streak of 10 blue eyed people in an ordered sample of 10 people is very rare, that is what you calculated. But getting a streak of 10 blue eyed people in an ordered sample of a million is not so rare. Also, getting a streak of 10 people who share some characteristic (among many) is not so rare even with substantially smaller samples.

But the chi squared test isn’t for that. It just answers the question if the raw probability of blue eyes in the PM population is different from the probability in the general population.
 
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Dale said:
It does actually. This is something that often trips people. Getting a streak of 10 blue eyed people in an ordered sample of 10 people is very rare, that is what you calculated. But getting a streak of 10 blue eyed people in an ordered sample of a million is not so rare. Also, getting a streak of 10 people who share some characteristic (among many) is not so rare even with substantially smaller samples.
Right, but the total run was not a million but 24. This we do know.
 
pinball1970 said:
They all definitely blue? How did you confirm? @DaveC426913
I checked every one individually. Yes, I trusted Google on this. Also, we know our PMs.
 
DaveC426913 said:
Right, but the total run was not a million but 24. This we do know.
OK, so for 10 out of 24 compared to an expected 6 out of 24 gives a p-value of 0.0593. That typically wouldn’t be considered statistically significant. But it is close enough that if some of those others had blue eyes then it would probably be significant.
 
Dale said:
OK, so for 10 out of 24 compared to an expected 6 out of 24 gives a p-value of 0.0593.
Consecutively?
 
DaveC426913 said:
Consecutively?
As I said above, the chi squared test doesn’t test that. It just tests if the sample proportion is the same as the expected proportion. There is a 0.0593 probability of getting 10 out of 24 blue eyed people if you expect 25 % (6 out of 24), just by random chance.
 
Dale said:
As I said above, the chi squared test doesn’t test that. It just tests if the sample proportion is the same as the expected proportion. There is a 0.0593 probability of getting 10 out of 24 blue eyed people if you expect 25 % (6 out of 24), just by random chance.
OK, so does that help us here?
 
DaveC426913 said:
OK, so does that help us here?
Yes. It lets us know that the 25 % is not terribly plausible (even assuming that none of the other 14 PM's had blue eyes)
 
A more Bayesian way to check that, starting with a uniform prior, 10 blue eyes out of 24 PMs gives a posterior distribution with a 95 % credible interval from 0.24 to 0.61 for the probability that a random PM has blue eyes.

Then, to see how "streaky" that is, I did a Monte Carlo simulation as follows:
1) draw a probability of blue eyes from the above posterior: ##p \sim Beta(11,15)##
2) draw 24 Bernoulli-distributed variables each with ##x \sim Bernoulli(p)##
3) count ##n## the longest run of 1's in the sample from step 2
4) repeat 1-3 for 1000 times

Out of 1000 trials 8 had a streak of at least 10 blue-eyed PM's. So it is not common, but at the few per thousand rather than the few per million level.

This accounts for the fact that blue eyes are more common among PM's than among the general population, and that the streak of 10 could occur with any 10 consecutive PM's (out of 24) not just the most recent 10.

It does not account for the fact that blue eyes are just one among many different physical characteristics that we would be surprised about. This is a big one, but it is hard to decide what to model for it.
 
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Dale said:
This accounts for the fact that blue eyes are more common among PM's than among the general population,
Can you explain the rationale that led this in small words?
 
DaveC426913 said:
Can you explain the rationale that led this in small words?
The observed 10/24 blue eyed PM’s is unlikely under the assumption that the probability of blue eyes is 0.25
 
I say that's pretty impressive. It's about as good as the time I saw ten people on the Japanese metro with only one on a smartphone.

My eyes turned bright blue for a year or so then switched back. I didn't know this was possible. Everybody who knew me from before noticed and remarked on it without any prompting so it wasn't subtle. At the time though, lacking Dave's research I failed to grasp the opportunity to become PM of Canada.

It's pretty well known that heads of state tend to be tall.
 
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DaveC426913 said:
Right, but the total run was not a million but 24. This we do know.
I count at least 2 with colour other than blue 'NOT Blue' , and some questionable ? due to artist rendering. ( did the atist choose an overly dark shade of because of lighting and angle )
MacDonald - blue
Mackenzie - blue
Abbott - blue
Thompson - blue
Bowell - blue
Tupper - NOT Blue
Laurier - ?
Borden - blue
Meighen - blue
Mackenzie King - ?
Bennett - blue
Louis St Laurent - NOT Blue
Diefenbaker - blue
Pearson - ?>> can't tell
Pierre Elliot Trudeau - blue
...

If by chance Pierson has blue eyes, then the run is 12.
Overal, it seems of the 24, at least 19 had/have blue eyes.
With a possible maximum of 22/24.

Would it be strange that all the names reflect mostly British ancestry for the ruling 'class'.
Northern Europe and eastern Slavic areas is a majority of blue eyed. For example, Iceland is 90% blue eyed. Britain +50% as a whole, with pockets greater than that. What is the % blue eyed from Ireland and Scotland?

[PDF] Prime Ministers' Portraits - publications .gc .ca​

publications.gc.ca › collection_2014 › parl › X9-28-2014-eng
https://www.google.com/url?esrc=s&o...0QFnoECAkQAg&usg=AOvVaw2KeoiIJVZRzSIVNxYCNSyM
https://www.google.com/url?esrc=s&o...0QFnoECAkQAg&usg=AOvVaw2KeoiIJVZRzSIVNxYCNSyM
 
That pretty much confirms that Canadians tend to vote for people with blue eyes. I wonder what the runners up had.
 
256bits said:
If by chance Pierson has blue eyes, then the run is 12. Overal, it seems of the 24, at least 19 had/have blue eyes. With a possible maximum of 22/24.
That changes things quite a bit.

With 19 blue-eyed PMs the posterior is ##p \sim Beta(20,6)## which has a 95 % credible interval from 0.59 to 0.91. Running that through the Monte Carlo simulation gives 364 out of 1000 runs with a maximum streak of at least 10 and 227 runs with a maximum streak of at least 12.

With 22 blue-eyed PMs the posterior is ##p \sim Beta(23,3)## which has a 95 % credible interval from 0.74 to 0.97. Running that through the Monte Carlo simulation gives 695 out of 1000 runs with a maximum streak of at least 10 and 549 runs with a maximum streak of at least 12.

So, what is happening here is not a highly improbable streak of blue eyes. The streak appears to be simply a consequence of the clear preference for blue-eyed PMs. Elections are not a random sample, and the Canadian PM sample is apparently highly biased towards blue eyes. This sample bias actually makes it quite likely to have a streak of at least 10 or 12 blue eyed PMs.
 
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Hornbein said:
My eyes turned bright blue for a year or so then switched back. I didn't know this was possible. Everybody who knew me from before noticed and remarked on it without any prompting so it wasn't subtle.
That's surprising. Maybe it's like the flamingos turning pink from eating algae and brine shrimp. ;-)
 
FactChecker said:
That's surprising. Maybe it's like the flamingos turning pink from eating algae and brine shrimp. ;-)
Canada PMs do like to snack on Blue Conservatives.
 
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Hornbein said:
That pretty much confirms that Canadians tend to vote for people with blue eyes. I wonder what the runners up had.
Silly me. The prime minister of Canada is appointed by the party with a plurality, not elected by public vote.
 
FactChecker said:
That's surprising. Maybe it's like the flamingos turning pink from eating algae and brine shrimp. ;-)
It was after I turned vegetarian. But then why did my eyes revert to their original color? A mystery of science.
 
Hornbein said:
It was after I turned vegetarian. But then why did my eyes revert to their original color? A mystery of science.
Did you get the blues from eating kale? Did you stop eating kale after you realized you would rather be dead? ;-)