I have a probability problem where I have to calculate the total probability and a Bayes probability from two events. The chances as given are:
$P(A) = 0.1, P(A^c) = 0.9$
$P(B|A) = 0.9, P(B|A^c) = 0.05$
Using these probabilities, I was able to work out the total probability:
$P(B) = P(A)P(B|A) + P(A^c)P(B|A^c) = (0.1*0.9) + (0.9*0.05) = 0.135$
And the chance of A given B using Bayes:
$P(A|B) = \frac{P(B|A)P(A)}{P(B)} = \frac{0.9*0.1}{0.135} = \frac{2}{3}$
Pretty standard stuff so far. However, the last part of the question adds an event C. The chance of C given A and B is 0.1. Knowing this, I have to calculate the chance of an outcome having A, B, and C. Is this really as easy as $P(A \cap B \cap C) = P(A)*P(B)*P(C) = 0.1*0.9*0.1 = 0.009?$
Use: $P(A \cap B \cap C) = P(C | A \cap B) P(A \cap B)$
Your answer assumes that the events $A, B, C$ are mutually independent which is not true. For example, $P(B) \ne P(B|A)$. In other words, knowledge of the event $A$ causes us to revise our probability of event $B$.