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...

## Comments