It’s always the little problems that create the biggest fuss, but in this case the Today programme’s debate on the chances of a carton of eggs all containing double yolks was unexpectedly illuminating.
Here’s the story. At about half past six, Sarah Montague and John Humphrys were debating the merits of a Daily Mail story which reported on a woman who had opened a carton of six eggs and discovered that every single one had a double yolk. The Mail went on to argue that since the chance of a single twin yolk is one in a thousand, the woman’s experience was a one-in-a-trillion event.
Sarah said that was right, multiplying a thousand by a thousand by a thousand, etc… John harumphed and said it couldn’t be true. And and hour later, they called me to referee.
This is much more interesting than it seems. If the double-yolk eggs appear with independent probabilities, then Sarah is right to multiply the thousands. The answer is one in a quintillion (or one in a trillion, if like the Daily Mail you prefer the traditional British trillion) – a number with 18 zeros. (Thinking about this some more I think the chance is actually one in six quintillion, since even when you get six double-yolked eggs, five times out of six they’ll be split across two consecutive cartons.)
On the other hand, John Humphrys was right to smell a rat – or a rotten egg. Because if the maths are correct, we are unlikely ever to have witnessed a carton of double-yolk eggs since the dawn of agriculture. And yet, several Today listeners have experienced just that.
So there are two thinking styles here. One is to solve the probability problem as posed. The other is to apply some common sense to figure out whether the probability problem makes any sense. We need both. Common sense can be misleading, but so can precise-sounding misspecifications of real world problems.
There are lessons here for the credit crunch. When the quants calculate that Goldman Sachs had seen 25 standard deviation events, several days in a row, we must conclude not that Goldman Sachs was unlucky, but that the models weren’t accurate depictions of reality.
One listener later solved the two-yolk problem. Apparently workers in egg-packing plants sort out twin-yolk eggs for themselves. If there are too many, they pack the leftovers into cartons. In other words, twin-yolk eggs cluster together. No wonder so many Today listeners have experienced bountiful cartons.
Mortgage backed securities experienced clustered losses in much the same unexpected way. If only more bankers had pondered the fable of the eggs.