Probability that no letter is in alphabetical order?

Question

Given a random string of distinct letters, find the probability that none of the letters are in order. A letter is in order when every letter preceding it is of lower alphabetical value and every letter after it is higher. having trouble with the combinatorial approach because there seems to be no easy way to avoid over-counting possibilities. Example of none in order: > dbac > > d: bac should precede it - not in order > > b: a should precede it and cd should come after - not in order > > a: bac should come after - not in order > > c: ab should precede it, d should come after it - not in order

Answer 1

You can find the probability that the NEXT character is not in alphabetical order.

a=1, b=2, c=3, d=4, ...., z=26

For any character ??? in the string, the probability that the next character

is NOT in alphabetical order is

( ???-1) / 26

For example, suppose the current character in the string is J=10.

then the probability that the next character is NOT in alphabetical order is (10-1)/26 = 9/26

So then you can multiply these probabilities of all but the last character in your string together

Note that if a is NOT the last character in your string, then the probability shall be zero

(since (1-1)/26 = 0 and there is no letter before a )