Another probability - random variables, generating functions

[Kate2010]

Homework Statement

When reading a book, you detect each mistake with probability p, independent of other mistakes. Let M denote the amount of mistakes on a certain page and D be the number that you detect on that page. Write down P[D=k|M=m] and find for k>=0 P[D=k].

Homework Equations

P(x=r) = (a^r)(e^-a)/(r!), poisson

The Attempt at a Solution

Earlier in the question I worked out for a textbook with n pages, number of mistakes on each page is poisson RV with parameter a, independent of mistakes on all other pages, that the expected number of pages with no mistakes is 1 - e^-a.

I tried using P[D=k|M=m] = P(D=k union M=m)/P(M=m) but don't know where to go from here.

 

[kurt.physics]

P[D=k] = P(D=k union M=m) + P(D=k union M=m+1) + ... I'm not sure how to simplify this expression. Any help would be much appreciated!