1. Real-life systems often are composed of several components. When the system's components are connected in parallel, the system works if at least one of the components is functional. When the system's components are connected in series, the system works if all of the components are functional.Consider the following network:

Let A¡, i=1,2,3,4,5 denote the probability that component a; is functional.Now, suppose that the events A¡ have the fooling properties: The status of component a4 and a5 are independent of the status of component a₁, a2, and a3. A₁ and A₂ are independent. P(A₁)= P(A₂)=0.5 P(A3|(A₁ N A₂)) = 0.4 \text { - } P\left(A_{3} \mid\left(\overline{A_{1}} \cup \overline{A_{2}}\right)\right)=1 / 3 \text { - } A_{1} \cap A_{2} \text { and } \overline{A_{1} \cap A_{2}} \text { form a partition. } A4 and A5 form a partition. Let C be the event that the system works. a) Find P(A3) b) Write an expression for C in terms of unions and intersections of the Ai's (and their complement) and find P(C) c) Given that the C occurs find the probability that az is functional