The question is:
There are 3 ways a CEO can commit accounting fraud: (A) he can overstate revenue by booking fake orders, (B) he can overstate profits by recording near-term expenses as longterm capital expenses, and (C) he can understate liabilities by making overly optimistic assumptions about the growth prospects of the company’s pension funds. In the United States, 50% of CEOs commit fraud (A), 20% of CEOs commit fraud (B), 15% of CEOs commit both fraud (A) and fraud (B), and all CEOs commit fraud (C). The probability of getting caught if a CEO does (A) is 0.01, independent of whether or not he commits other kinds of fraud. Similarly, the probability of getting caught if he does (B) is 0.05, and of getting caught if he does (C) is 0.001.
a) What are the chances (rounded to 2 decimal places) that a CEO who perpetrates all 3 kinds of fraud will get caught?
I feel like this is supposed to be a very simple question but I am struggling with it despite having an easy time with what I assumed were harder questions in the same place I found this one.
The problem I keep running into is that, any solution I can come up with requires I figure out the P(caught ∩ A ∩ B ∩ C). I have not been able to figure out P(caught ∩ A ∩ B ∩ C) so either I am looking in the wrong direction or there is a trick to solving for it that has not yet hit me.
Let $A, B, C$ denote committing crimes and $A_0,B_0,C_0$ denote caught with crimes.
a) What are the chances (rounded to 2 decimal places) that a CEO who perpetrates all 3 kinds of fraud will get caught?
In the language of my events, this question is asking for $$ P(A_0\cup B_0\cup C_0 |ABC). $$ The words "who perpetrates" is telling us the given and "caught" is a vague word for caught with one of the crimes.
By complement law, we have $$ P(A_0\cup B_0\cup C_0 |ABC)=1-P(A_0^c B_0^c C_0^c|ABC). $$ Now the trick is that regardless of what crimes were committed, it's always the case that the process of getting caught is independent. Hence, above equals $$ 1-P(A_0^c|ABC)P(B_0^c|ABC)P(C_0^c|ABC). $$ It's also the case that knowing other crimes were committed is irrelevant so this simplifies to $$ 1-P(A_0^c|A)P(B_0^c|B)P(C_0^c|C). $$