Re: help with simple probabilities

I'm no expert statistician, and can't solve the problem either. I
strongly suspect that there's a distribution for this sort of problem
that I either don't know or haven't recognised.

But, I think the following reasoning might possibly be heading in the
direction of an answer. Though of course I could easily be wrong.

Let's say that we are going to choose K letters. The chance of getting
all 'a's is 1/26 to the power of K. Since there are 26 letters to
choose from, there are (26 choose 1) = 26 possible single letter
sequences. Hence the probability of choosing a sequence of K letters
all the same is:

(26 choose 1) * (1/26)^K

Now I want to calculate the probability of a sequence of length K which
has *at most* two distinct letters. If the two letters were 'a' and
'b', then the probability of a sequence that has only 'a' and 'b' in
any combination (including either only 'a's or only 'b's) is (2/26)^K.
Since there are (26 choose 2) ways of choosing two letters from the
alphabet, the probability of getting a sequence that uses at most two
letters is:

(26 choose 2) * (2/26)^K

This is the probability of generating either a two letter sequence or a
one letter sequence. But, we know the probability of a one letter
sequence already, so all we have to do is subtract the probability of a
sequence containing a single letter, and we have the probability of
generating a sequence containing two letters.

This should easily generalise for any n. It's simple to replace 2 with
n in the above expression (note that n <= 26) which gives us the
probability of a sequence with at most n distinct letters. We then
subtract the probability of a sequence with at most n-1 distinct
letters, and that might be an answer. Note that this won't work for
n=1, but we have the n=1 formula above.

This is unlikely to be the right answer, but at least I had fun
thinking about it.

Cheers,

Ross-c

.

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