The Flaw In Littlewood’s Law

Introduction

Littlewood’s law is somewhat tongue in cheek, and was coined by Cambridge Professor John Littlewood in 1986. He defined a miracle as an exceptional event of special significance occurring at a frequency of one in a million. By assuming that the average person is alert for eight hours a day, with one event happening each second, one would expect to see a miraculous event every 35 days. The calculation being =1,000,000 odds / 60 minutes / 60 seconds / 8 hours.

This reasoning seems quite valid until we realise that one-in-a-million is an arbitrary probability so should not be given any special consideration over other probabilities. When we consider all other arbitrary probabilities and apply the same logic, we quickly run into problems.

Rival Law #1

Let us choose another arbitrary probability: one-in-one.

If we call something that happens one-in-one times a “definite” event, then by the same reasoning as Littlewood’s law we can claim that: One would expect to see a “definite” event of special significance each second. 

Rival Law #2

Let us now choose another arbitrary probability: one-in-two

If we call something that happens one-in-two times a “50/50” event, then by the same reasoning as Littlewood’s law we can claim that: One would expect to see a “50/50” event of special significance every 2 seconds. 

We could continue stating rival laws (one-in-three, one-in-four, one-in-five etc)… ad infinitum.

The Result

As one-in-a-million is a completely arbitrary probability, it should not be given any special consideration compared to other probabilities. So if we assume Littlewood’s law to be true we should equally assume rival laws 1, 2, 3 etc to also (and equally) be true.

Littlewood’s law works on the assumption that we experience one event per second. If all laws were assumed to be true (i.e. we should expect one-in-one events every second, one-in-two events, every two seconds etc) and if we only experience one event per second, then not all laws can actually be true. 

In the absence of any reason to believe that one-in-a-million events have any special status over one-in-one events or one-in-two events etc, we have no reason to believe that Littlewood’s law is actually true and would instead hold the contradictory expectation that miracles will occur every month and also not occur every month. 

Conclusion

Littlewood’s law may appear to be a good reason to discredit miraculous events, however as shown above, if we expect to see a miracle every month, we should also (and equally) expect not to see a miracle every month to avoid special pleading.  

We also see a contradiction between Littlewood’s law and the argument that “extraordinary claims require extraordinary evidence”.