My wife grew up on a chicken farm. When the chickens were young as you reported there were a good many double yolkers. Her Dad would keep them for the family because he liked the yolk. I would have many a double yolk omelet when we were dating.
Valid statistics depends on random samples. Once you inject human perception into the sampling, abandon all hope ye who enter here.
I enjoyed your article. For my entire life, my wife has insisted on jumbo eggs. So, my perception is that double-yolks are quite common. And they are, because of the selection process that precedes my experience with the eggs.
You have to wonder how our perception of many things in our lives is biased by a selection process that we never see or are not even aware of.
Among my circle of potluck acquaintances I am the "Devilled Egg Queen" and I always use jumbo eggs for these, although I buy large for ordinary cooking and baking because recipes are generally calibrated for those. So I knew exactly where you were going with the double yolk thing!
I love the Monty Hall Problem. I first studied statistics when it was a relatively new discussion. Then, some years later, I loved how Marilyn Vos Savant lured so many academicians into making chimpanzees of themselves—flapping their credentials about in the wind. Mark Haddon used the problem to great effect in his novel “The Curious Incident of the Dog in the Night-time” about an autistic kid who is fascinated by the problem.
As a "numbers guy" I loved this piece. I get very agitated at statements made with certainty about results which are decidedly uncertain. This is prevalent in the field of investing. I'm constantly shouting inside my brain "You can'y say that!" You've reminded me of a few items that never seem far away from usage:
1) Death and taxes...there's a reason that phrase sticks around
2) Risk means more things can happen than will happen - Elroy Dimson
3) We traditionally view lottery winning as good, but I often tell people there are negative lotteries (e.g. plane crash)
4) People (me too) complain about auto insurance rates going up, yet within 5 minutes yesterday I observed two different drivers in my rear view mirror with heads pointed 90 degrees down at their cell phones. Upon a stoplight turning green, one of them did not move for 8-10 seconds. Insurance is a risk probability x loss severity business. In automobiles distractions increase risk.
Finally, I remember a famous 1984 Warren Buffett article on investment performance, which utilized an entertaining introduction about consecutive coin flipping:
As an engineer and a mathematician I am regularly appalled at the statistical ignorance of the media and the ordinary citizens in general. When I see some probability quoted in the public press I immediately doubt the rest of the article.
Decades ago, I thought of creating a tool for journalists to do relatively simple statistical analyses to bolster their reporting. I decided against doing so because I decided that it would be equivalent to producing a whiskey-and-car-keys tool for teenagers.
I once had a carton of jumbo eggs and had the same experience with the double yolks. Thought it was odd. Now I get why that happened. Interesting article. Statistics may be a thing, 1 in 20 women will get breast cancer in their lifetime except until you get breast cancer then it’s 100% chance that you have cancer.
This is a great article. However, your probability calculation is incorrect.
The eggs labelled "Jumbo" are really a combination of two different kinds of eggs: "Real Jumbo" eggs and "Super Jumbo" eggs. When you purchase a carton of so-called "Jumbo" eggs, they are a random sample from this mixture distribution. Let's assume that the conditional probability of a double yolk, given that an egg is a "Real Jumbo", is 1/1000, and the conditional probability of a double yolk, given that an egg is a "Super Jumbo", is 1/2. Then the unconditional probability that a random egg drawn from a "Jumbo" carton has a double yolk will depend on the proportion of eggs that are really "Super Jumbo". For instance, if 90% of the eggs are "Real Jumbo" and 10% of the eggs are "Super Jumbo", then the probability of getting a double yolk egg is:
$$
\Pr(X = 1) = 0.9 * (1/1000) + 0.1 * (1/2) = 5.1%
$$
However, if 50% of the eggs are "Real Jumbo" and 50% of the eggs are "Super Jumbo", then the probability of getting a double yolk is:
$$
\Pr(X = 1) = 0.5 * (1/1000) + 0.5 * (1/2) = 25.5%
$$
So the unconditional probability of getting a double-yolk egg depends on the relative proportions of "Real Jumbo" eggs and "Super Jumbo" eggs. But even in the case where 50% of the eggs are "Super Jumbo" eggs, the probability of a double-yolk egg is at most about 1/4.
Thanks. I think the wording is subject to multiple interpretations. But my experience with this particular carton suggests that mine is correct. First of all, every egg in the carton is enormous, so I think they’re 100% super-jumbos. Secondly, we have now been through seven straight eggs and all seven have been double yolks. By Bayesian inference, the possibility that your math is correct is getting pretty dicey. But you’re not the only one who offered me this alternative interpretation. So thanks!
I stumbled on this article, and I feel like it implemented all the tips in articles on how to write a Substack article I've ever read. This is peak Substack-ing.
Another example from medicine. Someone comes to the doctor presenting with a set of symptoms. The doctor tests for the most likely explanation, but the test is negative. He tests for the next most likely explanation — also negative. After a few more iterations, the patient asks whether she might have disease X. The doctor scoffs “Oh, that’s a one-in-a million chance” and dismisses the idea.
Just because the disease is rare in the general population doesn’t mean it’s unlikely for this patient. Remember, you’ve already ruled out the most likely diagnoses. Whatever she has is bound to be unusual!
But it gets worse. How do we know the base rate is really one-in-a-million? If many doctors refuse to test for disease X because they believe it’s so rare, then this becomes a self-fulfilling prophecy.
Perhaps even worse—a doctor says there’s a million-to-one chance that you DON’T have some horrible illness, when, in fact, it’s pretty likely that you don’t. I wrote about a celebrated such example here: https://graboyes.substack.com/p/experts-with-statistics/. “In 1989, after Leonard Mlodinow received a positive HIV test, a doctor told him he had a 99.9% chance of dying of AIDS within 10 years. He didn’t have AIDS, never faced such grim probabilities, and is still around.” Long ago, I heard of some guy who killed himself after receiving such a statistical misdiagnosis. I haven’t been able to verify that that story is true.
This might be the most memorable piece I've read of yours, not least because my wife and I also cracked open 6 double-yolk eggs in a row just a few weeks ago.
Regarding Mr. & Mrs. Clark, the couple convicted in the SIDS deaths: given the timing, I have a strong feeling that the people in their trial wanted to avoid another Waneta Hoyt case.
Hoyt killed 5 of her kids in the 1960s and 1970s, deaths which were attributed to SIDS (including one kid who was 2 years old at the time of his death). She confessed in 1994 and was tried & convicted in 1995, and the coverage of her trial would likely have been fresh in the memory of law enforcement at the time of the Clark children's deaths.
I'll just call it the Rat problem, or the Roach problem.
Statistically, you're rather unlikely to see a mouse or roach in your home. They take up the least amount of space compared to any other living thing in the house or apartment, thus shrinking their statistical potential.
But, the moment you see one, rest assured, there are several.
My wife grew up on a chicken farm. When the chickens were young as you reported there were a good many double yolkers. Her Dad would keep them for the family because he liked the yolk. I would have many a double yolk omelet when we were dating.
Valid statistics depends on random samples. Once you inject human perception into the sampling, abandon all hope ye who enter here.
I enjoyed your article. For my entire life, my wife has insisted on jumbo eggs. So, my perception is that double-yolks are quite common. And they are, because of the selection process that precedes my experience with the eggs.
You have to wonder how our perception of many things in our lives is biased by a selection process that we never see or are not even aware of.
Among my circle of potluck acquaintances I am the "Devilled Egg Queen" and I always use jumbo eggs for these, although I buy large for ordinary cooking and baking because recipes are generally calibrated for those. So I knew exactly where you were going with the double yolk thing!
:)
Wonderful discussion! It reminds me of the Monty Hall problem discussion, though they are not the same
I love the Monty Hall Problem. I first studied statistics when it was a relatively new discussion. Then, some years later, I loved how Marilyn Vos Savant lured so many academicians into making chimpanzees of themselves—flapping their credentials about in the wind. Mark Haddon used the problem to great effect in his novel “The Curious Incident of the Dog in the Night-time” about an autistic kid who is fascinated by the problem.
As a "numbers guy" I loved this piece. I get very agitated at statements made with certainty about results which are decidedly uncertain. This is prevalent in the field of investing. I'm constantly shouting inside my brain "You can'y say that!" You've reminded me of a few items that never seem far away from usage:
1) Death and taxes...there's a reason that phrase sticks around
2) Risk means more things can happen than will happen - Elroy Dimson
3) We traditionally view lottery winning as good, but I often tell people there are negative lotteries (e.g. plane crash)
4) People (me too) complain about auto insurance rates going up, yet within 5 minutes yesterday I observed two different drivers in my rear view mirror with heads pointed 90 degrees down at their cell phones. Upon a stoplight turning green, one of them did not move for 8-10 seconds. Insurance is a risk probability x loss severity business. In automobiles distractions increase risk.
Finally, I remember a famous 1984 Warren Buffett article on investment performance, which utilized an entertaining introduction about consecutive coin flipping:
http://csinvesting.org/wp-content/uploads/2015/01/The-Superinvestors-of-Graham-and-Doddsville-by-Warren-Buffett.pdf
Great stuff!
How very interesting. And I hate math. as Disraeli said there are 3 kinds of lies, lies, damned lies and statistics.
Math can be great fun!
Now this brings back some old, unused memories: “chrono-synclastic infundibulum” in The Sirens of Titan. I thought it was de novo from Vonnegut.
As an engineer and a mathematician I am regularly appalled at the statistical ignorance of the media and the ordinary citizens in general. When I see some probability quoted in the public press I immediately doubt the rest of the article.
Decades ago, I thought of creating a tool for journalists to do relatively simple statistical analyses to bolster their reporting. I decided against doing so because I decided that it would be equivalent to producing a whiskey-and-car-keys tool for teenagers.
As somone who used have a home flock of about 30 chickens, I can say that double-yolks are not that rare. Maybe it's a breed thing as well?
Our carton is now seven straight. Five more to go.
I once had a carton of jumbo eggs and had the same experience with the double yolks. Thought it was odd. Now I get why that happened. Interesting article. Statistics may be a thing, 1 in 20 women will get breast cancer in their lifetime except until you get breast cancer then it’s 100% chance that you have cancer.
This is a great article. However, your probability calculation is incorrect.
The eggs labelled "Jumbo" are really a combination of two different kinds of eggs: "Real Jumbo" eggs and "Super Jumbo" eggs. When you purchase a carton of so-called "Jumbo" eggs, they are a random sample from this mixture distribution. Let's assume that the conditional probability of a double yolk, given that an egg is a "Real Jumbo", is 1/1000, and the conditional probability of a double yolk, given that an egg is a "Super Jumbo", is 1/2. Then the unconditional probability that a random egg drawn from a "Jumbo" carton has a double yolk will depend on the proportion of eggs that are really "Super Jumbo". For instance, if 90% of the eggs are "Real Jumbo" and 10% of the eggs are "Super Jumbo", then the probability of getting a double yolk egg is:
$$
\Pr(X = 1) = 0.9 * (1/1000) + 0.1 * (1/2) = 5.1%
$$
However, if 50% of the eggs are "Real Jumbo" and 50% of the eggs are "Super Jumbo", then the probability of getting a double yolk is:
$$
\Pr(X = 1) = 0.5 * (1/1000) + 0.5 * (1/2) = 25.5%
$$
So the unconditional probability of getting a double-yolk egg depends on the relative proportions of "Real Jumbo" eggs and "Super Jumbo" eggs. But even in the case where 50% of the eggs are "Super Jumbo" eggs, the probability of a double-yolk egg is at most about 1/4.
Thanks. I think the wording is subject to multiple interpretations. But my experience with this particular carton suggests that mine is correct. First of all, every egg in the carton is enormous, so I think they’re 100% super-jumbos. Secondly, we have now been through seven straight eggs and all seven have been double yolks. By Bayesian inference, the possibility that your math is correct is getting pretty dicey. But you’re not the only one who offered me this alternative interpretation. So thanks!
I stumbled on this article, and I feel like it implemented all the tips in articles on how to write a Substack article I've ever read. This is peak Substack-ing.
This is one of the nicest compliments I’ve ever received on my writing. I showed it to my wife, and your words have uplifted me all week. :)
Another example from medicine. Someone comes to the doctor presenting with a set of symptoms. The doctor tests for the most likely explanation, but the test is negative. He tests for the next most likely explanation — also negative. After a few more iterations, the patient asks whether she might have disease X. The doctor scoffs “Oh, that’s a one-in-a million chance” and dismisses the idea.
Just because the disease is rare in the general population doesn’t mean it’s unlikely for this patient. Remember, you’ve already ruled out the most likely diagnoses. Whatever she has is bound to be unusual!
But it gets worse. How do we know the base rate is really one-in-a-million? If many doctors refuse to test for disease X because they believe it’s so rare, then this becomes a self-fulfilling prophecy.
Perhaps even worse—a doctor says there’s a million-to-one chance that you DON’T have some horrible illness, when, in fact, it’s pretty likely that you don’t. I wrote about a celebrated such example here: https://graboyes.substack.com/p/experts-with-statistics/. “In 1989, after Leonard Mlodinow received a positive HIV test, a doctor told him he had a 99.9% chance of dying of AIDS within 10 years. He didn’t have AIDS, never faced such grim probabilities, and is still around.” Long ago, I heard of some guy who killed himself after receiving such a statistical misdiagnosis. I haven’t been able to verify that that story is true.
83% of statistics are simply invented.
83.6%
This might be the most memorable piece I've read of yours, not least because my wife and I also cracked open 6 double-yolk eggs in a row just a few weeks ago.
Regarding Mr. & Mrs. Clark, the couple convicted in the SIDS deaths: given the timing, I have a strong feeling that the people in their trial wanted to avoid another Waneta Hoyt case.
Hoyt killed 5 of her kids in the 1960s and 1970s, deaths which were attributed to SIDS (including one kid who was 2 years old at the time of his death). She confessed in 1994 and was tried & convicted in 1995, and the coverage of her trial would likely have been fresh in the memory of law enforcement at the time of the Clark children's deaths.
I'll just call it the Rat problem, or the Roach problem.
Statistically, you're rather unlikely to see a mouse or roach in your home. They take up the least amount of space compared to any other living thing in the house or apartment, thus shrinking their statistical potential.
But, the moment you see one, rest assured, there are several.