Solutions - 5020-03

(1) State the inclusion-exclusion principle:

$$P[A\cup B]=P[A]+P[B]-P[A\cap B]$$

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(2) Derive the lower bound on $P[AB]$:

Since $P[A\cup B]\le 1$:

$$\begin{array}{c}P[A\cap B]=P[A]+P[B]-P[A\cup B]\\ \\ \gg \gg P[A\cap B]\ge 0.4+0.7-1\gg \gg P[AB]\ge 0.1\end{array}$$

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(3) Derive the upper bound on $P[AB]$:

Since $A\cap B\subseteq A$, we have $P[A\cap B]\le P[A]$.

$$P[AB]\le 0.4$$