(Success of an event) Reliability analysis works not only in finding how the whole system is dependent on its multiple components, but makes sense in checking whether the desired event with several...

(Success of an event) Reliability analysis works not only in finding how the whole system is dependent on its multiple components, but makes sense in checking whether the desired event with several causes/ factors would happen. Suppose you want to learn how to bake during this quarantine, and the three important factors of a successful baking are ingredients, temperature and time. The ideal ingredients for the flour is cake flour, and you can also substitute it with a mixture of all-purpose flour and corn starch with certain ratio. Suppose you have sufficient cake flour w.p. p1, and sufficient all purpose flour and sufficient corn starch for the substitute mixture w.p. p2 and p3. After successfully preparing the ingredients, you hope to set up the right temperate and cook time for the oven. However, unfortunately there is something wrong with the oven’s electronic display. Therefore w.p. p4 you set the temperature correctly and w.p. p5 you set the time appropriately.

1. 
   

   (a)  Let A denote the event that you successfully bake the cake. Write A as a function of sub events de- scribed above by composing nested unions and intersections. Clearly define each of the subevents. (Hint: There should be 5 sub-events A1, A2, . . . A5.)

   
   


2. 
   

   (b)  Define x1, x2, . . . x5 to be the indicator functions whether the events A1, A2, . . . A5 are true, respec- tively. Write down the structure function φ(x1 , . . . x5 ) in terms of x′i s as defined, which outputs 1 if event A is true, i.e. you successfully baked the cake, and 0 otherwise. You do not need to simplify the results, but do get rid of the max/ min functions.

   
   


3. 
   

   (c)  Assume all the sub-events are independent. Compute the probability of event A as a function of p1,p2,...p5.

   
   


4. 
   

   (d)  Note the independence assumption in the above problem may not be true. For example it may be that many recipes use combinations of cake flour, all purpose flour, and corn starch such that the amount left in your pantry of these items tend to be dependent. Which part of the derivation of your above computed probability fails?

   
   


5. 
   

   (e)  Can you think of ONE case/ circumstance when the approximations P(min(xi,xj)) ≈ pipj and P(max(xi,xj))≈(1−pi)(1−pj)≈pipj may be nearly true even without independence?