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1. (Set Theury asd Arwems of Predabihty) (25 pts) (a) (Topts) Let events \( A_{1} \) and \( A_{2} \) watisfy \( P\left[A_{1} \mid=1\right. \) and \( P\left|A_{2}\right|=1 \). Show that \( P\left(A_{1} \cap A_{2} \mid=1\right. \). Therruchums For this quetion you are not allowed to use Venn diapratms. (b) \( (S p t s) \) Suppose that \( P\left[A_{1} \cap A_{2}\right)=1 \), Show that \[ P\left[A_{2}\right] \geqslant \frac{1}{2} \] Irsetracthons: You are allowed to use any tools at your disposial. (c) (Spts) Suppore that \( P\left[A_{1}\right]=P\left[A_{2}\right]=\frac{1}{2} \), and \( A_{1} \) and \( A_{2} \) are independent, Find \( P\left[A_{1} \cup A_{2}\right] \). Instructions: Show all the steps. (d) (5 pts) Here need to justify by clearly shourng your steps. In an experiment, \( A \) and \( B \) are independent events and we know that \( P[A \cap B]=\frac{1}{3} \) and \( P[A]=\frac{1}{2} \). Are \( A \) and \( B^{\prime} \) mutually exclusive?