The Podcast has been restored! Sorry for the mix up yesterday.
Here’s the story cut short. A woman was browsing a used book stall and came across a volume she owned as a child. The Wall Street Journal asks:
What are the chances? For most of us this is a rhetorical question, expressing our surprise at such a seemingly magical coincidence. But Joseph Mazur, the author of “Fluke: The Math and Myth of Coincidence,” has a more precise answer. The odds that Anne Parrish would stumble upon her book years later in Paris are 3,331 to 1 or, as he puts it, “slightly better than the odds of being dealt a poker hand of four of a kind.” Not so magical, then, after all…
Mazur assigns a probability to each link in the chain of events: the likelihood that Anne would visit Paris in 1929 (0.1, or 10%); the likelihood that she would browse the bookstands (0.3); the likelihood that the book would be there (0.01). The chances that all three would take place and that Anne discover her childhood book are therefore 0.1 x 0.3 x 0.01 = 0.0003, or 3,331 to 1.
This won’t do. Why? Consider this: Briggs assigns a probability to each link: the chance Anne visits the book dealer is 0.987654321; and 0 likelihoods for the rest. Chances she gets her own book are therefore 0.987654321 x 0 x 0 = 0. Hey. Why not? I am a full qualified statistician, holding the coveted posthole digger award.
This means, given my superior prowess at assigning odds, that Anne never got her book, even though she said she did. An investigation is in order.
How did Mazur find different probabilities? He made them up, just as I did. That’s because there are no probabilities “for” this “event.” None, except one. I mean one, as in 1. The probability Anne found her book is 1, because why? Because she found her book. It happened. Conditional on the evidence provided in the story, the probability “Anne found her book” is 1.
Conditional on whatever evidence Mazur made up, the probability is 0.0003. And, to be pedantic, conditional on the evidence I made up, the probability is 0.
All of these probabilities are correct, but only one of them is true. To see that, consider also that the proposition “Anne would have found her book had she tried again” makes no sense, but would have to if we can find conditioning evidence upon which we could agree.
Change the story to this. Anne went to the bodega and bought a lottery ticket, which won. Then, conditional on the evidence of this new story, the probability of “Anne won” is 1. But the probability of “Anne will win” given the evidence she bought a ticket and evidence about the rules of the lottery, the probability can be calculated. That seems like the right answer, which it is but isn’t.
This is because, in both the case of Anne finding her book and winning the lottery, cause is involved. Some (enormous) list of things caused both Anne’s book and Anne to meet at that time and place (and for Anne to have enough money on her). And some list of things caused the lottery to register the numbers that matched Anne’s tickets and some things caused Anne to have those numbers printed on her ticket.
Even though we don’t know the details of the causes of the ticket, we have a fair idea of their nature. This is why we can agree on the evidence needed to form the probability of “Anne will win.” Better evidence exists, however, and that is knowledge of the causes, but we don’t have that (though somebody might, say a physicist in charge of the lottery machine).
We have next to no idea of the nature of the causes of Anne finding her book, which is why we can’t agree, nor should we, on the evidence that gives the probability of “Anne finds her book.”
Anne finding her book feels like a coincidence, yet her winning the lottery doesn’t. Why? Because of our lack of any knowledge of cause in the first instance, and knowledge of the nature of cause (but not precise details) in the second.
We all know that something caused Anne to find her book, but we have no idea what. So it’s hard not to think that some overarching cause guided the event. This might even be true. There is no evidence in the event itself one way or the other. Positing a “guide” is simplifying a complex situation, and that is pleasing in and of itself.
There’s much more to be said about coincidence—which I’ll leave for another day.
Thanks to John Watkins for alerting us to this topic.