TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Bura)/Übungen 2020W/HW02.1

1) Bonferronis’s inequality

1. Let ${\textstyle A}$ and ${\textstyle B}$ be two events and ${\textstyle P(A)=0.8}$ and ${\textstyle P(B)=0.7}$.
   Show that ${\textstyle P(A\cap B)\geq 0.5}$.

2. Prove that for any two events ${\textstyle A}$ and ${\textstyle B}$ Bonferroni’s inequality holds

   $$P(A\cap B)\geq P(A)+P(B)-1$$

Lösungsvorschlag von Friday

--Friday Sa 30 Jan 2021 14:19:46 CET

From the lecture we know that the following axioms are true.

$${\begin{aligned}P(A\cup B)&=P(A)+P(B)-P(A\cap B)\\P(A\cap B)&\leq 1\end{aligned}}$$

Part a)

$${\begin{aligned}P(A\cup B)&=P(A)+P(B)-P(A\cap B)\\P(A\cup B)&\leq 0.8+0.7-0.5\\P(A\cup B)&\leq 1\quad \Box \end{aligned}}$$

Part b)

$${\begin{aligned}P(A\cap B)=P(A)+P(B)-P(A\cup B)\geq P(A)+P(B)-1\quad \Box \end{aligned}}$$