How does inclusion exclusion relate to the complements

Question

Let $A_1,A_2,A_3,A_4$ be events such that for every $i,j,k=1,2,3,4$,

$P(A_i) = \frac{1}{2}$,

$P(A_i \cap A_j)= \frac{1}{3},\quad i\ne j$,

$P(A_i\cap A_j\cap A_k) = \frac{1}{4},\quad i\ne j, j\ne k, k\ne i$,

$P(A_1\cap A_2\cap A_3\cap A_4) = \frac{1}{5}$.

Find $P(A^c_1\cap A^c_2\cap A^c_3)$.

Progress: I have obtained $$P( A_1 \cup A_2 \cup A_3 \cup A_4) = \frac{4}{5}$$

Any clues on how I can find the intersection of the complements? Thanks!

Answer 1

Use De Morgan's law :

$$\bigcap \overline{A_i} = \overline{\left(\bigcup A_i \right)}$$

So $$P(A_1^c \cap A_2^c \cap A_3^c)=P(\overline{A_1 \cup A_2 \cup A_3})$$ $$=1-P(A_1 \cup A_2 \cup A_3)$$