Six Yolks from Three Eggs: What Are the Odds of That?

Making breakfast one recent morning, a colleague cracked an egg and BOOM! A double yolk came out. He cracked another. SHAZAM! And a third, BOOYAH! Three double yolks in a row, what are the odds of that?

It turns out, that number is easy to calculate: In general, one out of every thousand eggs is a double, which would calculate the odds at 1,000 x 1,000 x 1,000, or one in a billion. Before we all went out and splurged on a pack of scratch tickets, however, we delved a little deeper.

It turns out that doubles turn out more frequently among young hens than older birds, and that flocks of hens tend to be the same age. The chance of a young hen laying a double-yolked egg are roughly 1:30. So, three in a row would calculate the odds at one in 27,000.

So which is, one in a billion or one in 27,000? You can see how, if this were a business case, whichever assumption is in error could have enormous implications on your financial modeling, response rate forecasts or whatever you are trying to predict.

On reflection, both of these calculations seem to me to be erroneous. If, in general, one gets a double every 1,000, then it strikes me the odds of the first egg being a double is 1:1000. Once you get one, assuming the "all eggs in the carton came from a flock of young hens" rule, the chances of a second would be 1:30 and the third would be another 1:30. So, our best guess of the odds of getting three in a row are 1,000 x 30 x 30, which is one in 900,000.

Why is this important? Because probability, statistics and math are hugely important to so many of our business processes. And more often than you might think, folks distort the assumptions that go into statistical modeling for a wide range of reasons, be they political, emotional, or to align conclusions with an expectation of the outcome.

Science and math by their nature seem factual, to be taken at face value. We need to remember that most scientific modeling comes with built-in assumptions, and these must also be factual or the whole conclusion can be suspect.

In this small example, we might have pointed to the 1:1,000,000,000 odds to make the point that something truly extraordinary had just happened. Our friendly competitor might want to tear us down a notch and challenge our assumptions, coming up with the 1:27,000 "meh" number. Yet, a balanced calculation results in a more nuanced conclusion, with the most accurate prediction of the likelihood of a six-yolk breakfast from the crack of three eggs being somewhere just short a million to one.

If you want to challenge an argument, find the assumptions that are used to justify it and delve a little deeper. You'll be surprised at what you might find, and how your predictors might become more accurate.

Update: The next morning, he cracked three morning and got three more doubles! Now that's extraordinary!