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I have been attempting to solve this set theory question, and am unsure exactly how to answer part b) :

Set Theory Question

For part a), I got P[A] = 0.4 and P[B] = 0.5. For part b), the only way I can see to solve it is to assume that $P[A \cup B \cup C] = 1$. This is my current working:

$$ \begin{align} P[B^c] &= 1 - P[B] \\ \therefore P[B^c] &= 1 - 0.5 = 0.5 \\ \\ P[B^c] &= P[A\cup C] - P[B \cap C] - P[A \cap B] \\ \therefore 0.5 &= 0.7 - 0.2 - P[A \cap B] \\ \therefore P[A \cap B] &= 0 \end{align} $$

Am I correct in my assumption, or is this another way to solve b) without assuming anything?

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Your assumption is true. Note that \begin{align} P(A\cup B\cup C)&=P((A\cup C)\cup( B\cup C))=\\ &=P(A\cup C)+P(B\cup C)-P((A\cup C)\cap(B\cup C))=\\ &=0.7+0.7-P(C)=1 \end{align} The other calculations are correct.

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