# Standard deviation of a set of random odds

#### giacecco

##### New Member
Standard deviation of a set of random probabilities

Imagine I have n mutually exclusive events whose probabilities are perfectly random. The sum of their probabilities is 1 of course. What is their most likely standard deviation, if there is one?

Playing around with R, it looks like there is one. If n is 10, for example, the standard deviation tends to 0.05891 .

Code:
> gen <- function (n) { x <- runif(n, 0, 1); x <- x / sum(x); x; }
> mean(replicate(10000000, sd(gen(10))))
[1] 0.05891704
What I would like to understand is:

a) Is what I found a known property of random distributions of probability? Or it is an aberration caused by using software random number generators?

b) If the answer to (a) is yes, how is this phenomenon called, so that I can find out more? And in particular...

c) Is there an exact formula to calculate the ideal sd value as a function of the number of events?

Thanks,

Giacecco

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