Multiply, add, or condition on probability? Sometimes it's quite confusing when it comes to determining how to answer a probability question. Confusions always arise as of whether I should multiply/add or make conditional the probabilities. For example the following:

Consider influenza epidemics for two parent heterosexual families.
Suppose that the probability is 15% that at least one of the parents
has contracted the disease. The probability that the father has
contracted influenza is 10% while that the mother contracted the
disease is 9%. What is the probability that both contracted influenza
expressed as a whole number percentage?

Let P(F) = Probability that father catches it; P(M) for mother.
I thought the P(both catch it) = P(F)P(M), but the answer is P(at least 1 catch it)= P(F)+P(M)-P(F AND M) and solve for P(F AND M).
My first question is that: I find it particularly difficult to differentiate between addition or multiplication rule when it comes to probabilities from independent events.
My second question is that: I'm also thinking if I'm to use P(at least 1 catch it)= P(F)+P(M)-P(F AND M), I would have make something like: P(at least 1 catch it)= P(F)P(NOT M)+P(M)P(NOT F)+P(F AND M). But it seems the P(F AND M) from two cases are not equivalent? Aren't these 2 expressions representing the same thing?
My third question, even when I calculate P(at least 1 catch it) = 1-P(both not catching it) = 1-P(NOT F)*P(NOT M), P(at least 1 catch it) does not equal to .15 given in the question. What's wrong with my calculation?
Are there any rules in governing which approach to use when solving a probability problem?


My first question is that: I find it particularly difficult to differentiate between addition or multiplication rule when it comes to probabilities from independent events.

That's not a question (you don't ask anything), but the answer to what I assume is your implied question is simple: there isn't an addition rule for independent events.
The "addition rule" $P(A \text{ or } B) = P(A)+P(B)$ is for mutually  exclusive events.
Draw a Venn diagram, from which it's obvious why there's another term there for non-mutually exclusive events (representing the overlap which gets counted twice, once in A and once in B, whereupon you must then subtract one of the overlaps back off again).

My third question, even when I calculate P(at least 1 catch it) = 1-P(both not catching it) = 1-P(NOT F)*P(NOT M), P(at least 1 catch it) does not equal to .15 given in the question. What's wrong with my calculation?

Note that the multiplication rule requires independence.
Did you make sure the events whose probability you multiplied were independent?

Rules for union ("OR") and intersection ("AND") are:
(i) $P(A \text{ or } B) = P(A) + P(B) - P(A  \text{ and } B)$
(ii) $P(A \text{ and } B) = P(A)\times P(B|A)$ $\:$ (General product rule)
If you have mutually exclusive events, the third term on the RHS in (i) is $0$, whence "addition rule for mutually exclusive events".
If you have independent events, the second term on the RHS in (ii) is equal to $P(B)$, whence "multiplication rule for independent events".
