Notes from class regarding the question: how does the (log) probability of a string change if you add 10 e’s to the end? This exercise is supposed to help you answer the homework exercise.

probability of original string

probability after we add 10 e’s to it:

take the ratio of the two:

=

Get rid of denominators, and note which exponents sum to N: :

Flip about:

Cancel identical factors:

Separate a bit:

Simplify:

taking logs:

Or, in words: the change in log probability is the sum of three things:

the change in how each of the old e's probability changed, minus the cost of the ten new e's, minus the change in all of the letters' probability due to general "diluting" (increase in length of string).