So I have a practice question for spam filtering what I'm trying to solve.
Given information before the question:
(a) An incoming email message is 9 times more likely to be not flagged as a spam than to be flagged.
(b) An incoming email message is 19 times more likely to be genuine than to be a spam.
Questions:
(a) What is the probability that an incoming message is flagged as a spam and what is the probability that an income message is actually a spam?
(b) Assume that an income message is a spam. What is the smallest probability that it will be flagged as a spam?
(c) Assume that an income message is flagged as a spam. What is the largest probability that it indeed a spam?
If I'm correct about the given information it should look like this:
Did I get the "likelihood" correct? For question (a) is it correct to say incoming message P(Flagged = Spam) = 1/10 = 10% and how would I calculate the second part of question (a) if it is indeed a spam?
You have computed the probabilities of being flagged and being spam correctly, (1/10 and 1/20 respectively), but your diagram assumes that flagging and being spam are independent of each other, which is not necessarily the case from the information in the question alone.
Let $F$ be the event that the email is flagged, and $S$ be the event that it is spam. We know $P(F)=0.1$ and $P(S)=0.05$, but not that $P(F \cap S)=0.005$. Parts (b) and (c) of the question build on this idea and look at the possible ranges of $P(F|S)$ and $P(S|F)$ without further knowledge on how the spam filter works.