Solutions - 5020-01

(1) Apply inclusion-exclusion:

$$\begin{array}{c}P[A\cup B]=P[A]+P[B]-P[A\cap B]\\ \\ \gg \gg 0.4+0.3-0.1\gg \gg 0.6\end{array}$$

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(2) Compute complements:

$$\begin{array}{c}P\left[A^c\right]=1-P[A]\\ \\ \gg \gg 1-0.4\gg \gg 0.6\end{array}$$ $$\begin{array}{c}P\left[B^c\right]=1-P[B]\\ \\ \gg \gg 1-0.3\gg \gg 0.7\end{array}$$

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(3) Use $P[A]=P[A\cap B]+P[A\cap B^c]$ to find $P[A\cap B^c]$:

$$\begin{array}{c}P[A\cap B^c]=P[A]-P[A\cap B]\\ \\ \gg \gg 0.4-0.1\gg \gg 0.3\end{array}$$

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(4) Compute the complement of $A\cap B$:

$$\begin{array}{c}P\left[{(A\cap B)}^c\right]=1-P[A\cap B]\\ \\ \gg \gg 1-0.1\gg \gg 0.9\end{array}$$