1 A certain section of a library contains several thousand books. A lecturer is looking for a book that refers to a particular topic. The lecturer believes that one-twentieth of the books in that section of the library contain a reference to that topic. However, the lecturer does not know which books they might be, so the lecturer looks in each book in turn for a reference to the topic. The first book the lecturer finds that refers to the topic is the \(X\) th book in which the lecturer looks.

1. A student says, "There is a maximum value of \(X\) as there is only a finite number of books. So a geometric distribution cannot be a good model for \(X\)." Explain whether you agree with the student.
2. 1. State one modelling assumption (not involving the total number of books) needed for \(X\) to be modelled by a geometric distribution in this context.
   2. Suggest a reason why this assumption may not be valid in this context. Assume now that \(X\) can be well modelled by the distribution \(\operatorname { Geo } ( 0.05 )\).
3. The probability that the lecturer needs to look in no more than \(n\) books is greater than 0.9 . Find the smallest possible value of \(n\).
4. The lecturer needs to find four different books that refer to the topic. Find the probability that the lecturer wants to look in exactly 40 books.