Bookshop probability question On being offered a new book, a bookshop reckons that either 1% of its regular customers will purchase the book with probability 4%, or 3% will buy with probability 30%, or 5% will buy with probability 66%. Only the latter case is considered profitable. If a random survey of 100 customers shows 6 would buy, what is the probability of profit to the bookshop if it accepts the new book? [Answer = 0.869184]
I'm confused as to what to do with the random survey information do i raise (0.05)(0.66) to the power 6 or what? Please help.
 A: HINT: Ah!  The confusion seems to be about the word "book" -- meaning a title vs a copy.  :)  My interpretation is as follows.
Given any book (and really, this means a title like "Hamlet", not a particular copy), it is exactly one of 3 types:


*

*Event $A$: 1% of customers would buy a copy of this book.  $P(A) = 4$% is given.

*Event $B$: 3% of customers would buy a copy of this book.  $P(B) = 30$% is given.

*Event $C$: 5% of customers would buy a copy of this book.  $P(C) = 66$% is given.

*Note that $4 + 30 + 66 = 100$%, as required.  BTW, I found the question wording rather confusing until I noticed this, at which point I realized this is the prior and they mean for you to apply Bayes.
Anyway the prior is part one.  Now part two, the survey: It turns out (event $E$) out of 100 customers 6 of them said they would buy a copy.
You are asked to find $P(C|E)$.  


*

*By Bayes' theorem: $P(C|E) = P(E|C) P(C) / P(E)$.  You're already given $P(C)=66$%.

*Now how would you calculate $P(E|C)$?  Well, conditioned on the book being type $C$, every of the 100 surveyed people would independently have a 5% chance of saying they would buy a copy, so what is the prob that exactly 6 of 100 would say Yes?

*Similarly, $P(E) = P(E|A)P(A) + P(E|B)P(B) + P(E|C)P(C)$
Can you finish from here or do you need more hints?
