Question # 1: On the network which I posted above, I am having trouble determining what the probability of:
P(A,F)
is and how it is derived?
My thinking was that if you have this event (A) that is true, has occured, and this other event (F) that is the child of event (A). In this case the probability of event (F) happening is heavily dependent on whether or not event (A) has occured. Therefore, if both occur then you can find the probability of both happening by multiplying their truth values:
P(A) * P(F|A) = 0.8 * 0.8 = 0.64 --> 64%
Question # 2: Similiar to the question above how can I find the probability of:
P(M, not(C) | H)
is and how it is derived?
I did some additional research and came to the conclusion that P(M, not(C) | H)
translates into P(M | not(C), H) * P(not(C) | H).
So in english, the probability of the event (M) occuring given that event (C) did not occur, is false and the event (H) occured multiplied by the probability of event (C) not occuring, is false, given that event (H) occured.
Thus, the total probability of the first operand: P(M | not(C), H) = .04 --> 4%
From what I gathered is we are trying to find the probability of event (M) occuring given that event (C) did not occur and event (H) occured. Therefore, I multiplied .4, which is the probability of event (C) not occuring by the probability of event (H) occuring given the event (C) did not occur, which is .1.
.4 * .1 == .04 --> 4%
Thus, the probability of the second operand:
P(not(C) | H) can be expanded by Bayes Theorem =
(P(H | not(C)) * P(not(C)) / P(H))
(.1 * .4) / .7 = .05714 --> 5.714%
I got the .7 from the probability of event (H) happening regardless of the dependence on event (C) being true or false.
In conclusion,
.04 * .05714 = 2.2857*10^3 --> .002857 --> 0.2857 % (Less than 1 %)
Just need some clarification if both of these are correct, but more importantly if my logic of each problem is heading down the right road.