Probability of Mistakes in a Book: Finding P[D=k|M=m]

[Kate2010]

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].

I've 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), any ideas how else to go about this?

 

[lippyka]

P[D=k|M=m] = P[D=k ∩ M=m]/P[M=m] = (p^k)((1-p)^(m-k))/(mCk*(1-p)^m)For k>=0 P[D=k] = Summation from 0 to m of P[D=k|M=m]*P[M=m] = Summation from 0 to m of (p^k)((1-p)^(m-k))*(e^(-a)*(a^m))/(m!*(1-p)^m) = e^(-a)*(a^m)*Summation from 0 to m of (p^k)((1-p)^(m-k))/(m!*(1-p)^m)