Sunday, August 24, 2003

The Improbable is more Probable than You'd Think

This interesting little item in the Independent goes some way in showing just how hard it is for most of us to grasp the concept of randomness. When statisticians say that a class of events has a random distribution, or that the odds of a certain outcome occuring are extremely low, most of us seem to take it to mean something very different from what it ought to mean. For instance, the odds that are random sequence of numbers should go

1, 2, 3, 4, 5, 6, ...

is exactly the same as it being

1, 5, 3, 4, 8, 5, ...

i.e., 1 in a million, yet most of us would jump to the conclusion that a random number generator that spat out the first sequence was rigged! The same thing goes for lottery tickets - as long as the lottery winner is picked entirely at random, it makes the most sense to pick as "predictable" a number as one possibly can; the odds of winning are no better or worse for so doing, but the odd human tendency to avoid "predictable" sequences when buying lottery tickets means that the odds of having to share any jackpot will be much lower.

The article is certainly worth reading, if you're into curiousities. I did spot one mistake in it though: the article claims that having 23 people in a room is sufficient to give a 50 percent chance that two will share the same birthday (which is true enough), but then explains this by saying that 23 people gives 256 pairings. Apart from being utterly uninformative (the number of pairings has no significance unadorned of context*), the number given is simply wrong - it should be (23 * 22)/2, or 253.


*The real reason is that while the probability of being born on a particular day is roughly 1/365, the probability of all the other 22 people being born on different days is

(364/365)*(363/365)*....((343)/365)

or, in a more concise fashion,

p = 365!/((365^23)*342!)

The odds of any two sharing a birthday is then given by (1-p), which works out to approximately 0.507297...