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I know $P(A) = 0.5$, $P(B) = 0.2$, and $P(C) = 0.6$

A and C are independent, and B and C are mutually exclusive.

I am interested in the following four quantities:

  1. $P(A \cap B \cap C)$
  2. $P({A \cup B} | C)$
  3. $P(A \setminus C)$
  4. $P(A \cup B \cup C^{c}$)

I know the values are respectively, 0, 0.5, 0.2, 0.7. But how do I find these given the information I know?

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    $\begingroup$ Let's look at the first. $B$ and $C$ are mutually exclusive so... what doe sthat mean about $B\cap C$? What does that mean about $A\cap B\cap C = A\cap (B\cap C)$? $\endgroup$ – JMoravitz Oct 6 '19 at 16:26
  • $\begingroup$ Since B, C are mutually exclusive, then $B \cap C$ is the empty set, so $P(A \cap \emptyset) = 0$. Thanks! $\endgroup$ – user581882 Oct 6 '19 at 16:38
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    $\begingroup$ Good, now, $P(A\setminus C) = P(A)-P(A\cap C)$, what do you know about $P(A\cap C)$ since $A$ and $C$ are independent? $\endgroup$ – JMoravitz Oct 6 '19 at 16:43
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    $\begingroup$ You seem to be doing fine then. What specific difficulties are having with the remaining two? Just confidence? $\endgroup$ – JMoravitz Oct 6 '19 at 16:48
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    $\begingroup$ $P(A\cup B\mid C) = \dfrac{P((A\cup B)\cap C)}{P(C)}=\dfrac{P((A\cap C)\cup (B\cap C)}{P(C)}=\dots$ $\endgroup$ – JMoravitz Oct 6 '19 at 16:54
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  1. $$ P(A\cap B\cap C)=P(A\cap (B\cap C)) = P(A\cap \phi)=P(\phi)=0 $$
  2. $$ P(A\cup B | C) = P(A|C) + P(B|C)-P(A\cap B|C)=P(A|C)=P(A)=0.5 $$ The first transition follows from the fact that $P(\cdot|C) $ is a probability function and regular rules hold, the second follows from the fact that $B,C$ and $A\cap B,C$ are mutually exclusive.
  3. $$ P(A \backslash C) = P(A)-P(A\cap C)=P(A)-P(A)P(C)=0.5-0.5\times0.6=0.2 $$
  4. $$ P(A\cup B\cup \overline{C})=P(A)+P(B\cup \overline{C})- P(A\cap(B\cup \overline{C}))$$ $C$ and $B$ are mutually exculsive, that means that for every $\omega\in B:\omega \notin C\rightarrow\omega\in \overline{C}$, so $B\subseteq \overline{C}$.
    $$ P(A\cup B\cup\overline{C})=P(A)+P(\overline{C})-(A\cap \overline{C})=P(A)+P(\overline{C})-P(A)P(\overline{C})=0.5+0.4-0.5\times 0.4=0.7$$
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