## 23 December 2009

### Unlikely Events

In his book The Ancestor's Tale, Richard Dawkins discusses how New World monkeys, who it is believed are descended from Old World monkeys, managed to get to South America. At the time this must have happened, South America had detached itself from Africa and was not yet joined to North America, so was effectively a large island. It is conjectured that rafting may have been how they got there: one or more monkeys ending up washed out to sea on floating trees for example. (If it were one it would have to have been a pregnant female.) Some people object to this hypothesis because they believe that this event is so unlikely it could not have happened, but the truth is that unlikely events happen all the time. Evolution Theory relies on the fact that extremely rare events (favourable mutations) do occur if you wait a long time - and evolution works over many millions of years.

After reading Dawkin's account I decided to see if I could come up with a mathematical example that illustrated this point. If the chance of a viable breeding group of monkeys getting to South America in any one year is, say, 1 in a million then we could calculate the chance of this happening if we wait, say, a million years. (This event would have occurred somewhere between 25 and 40 million years ago so waiting a million years is reasonable.) It is actually easier to calculate the chance of failure to get across and subtracting that from 1 to get the answer. I did the calculation for a probability of 1/n of success in one year, which gives (1 - 1/n) as the probability for failure. Because the events are independent, the probability of failure after 2 years is that multiplied by itself or
(1-1n)2 . For n years this is
(1-1n)n , which for n=1 million gives 0.368 to 3 significant figures, or 0.632 as the probability of success. If you tried with the same probability for 2 million years then the chance of success would be 0.865, and if you kept on trying for 10 million years it would be 0.9999, or anything but unlikely.

The formula can be generalised by adding another variable, so
1 - (1-1n)rn would give the probability of success for 2 million years if r=2 and n=1 million. For values of n greater than 1 million the probability is the same to 3 significant figures so it is tempting to take the limit of the formula as n goes to infinity:
Lim
n→∞
1 - (1-1n)rn = 1 - e-r where e is approximately 2.71828. Note that this implies that an event with zero probability will still happen if you try it an infinite number of times!

Of course this should not be taken too seriously: the probability of one in a million is pure conjecture, and it would not be the same for every year. Over millions of years as the continents drifted further apart, the chances of a successful crossing would go down.

Leaving aside this particular example there are, in general, an infinite number of unlikely events that may happen. If you take a million possible unlikely (and unrelated) events, each with a chance of one in a million of happening in a given time period, then the maths shows that there is a good chance that one of these events will actually happen. People tend to notice only the seemingly unlikely events that do happen, and some times start to look for mystical explanations when usually no special explanation is required.