A system is composed of 5 components, each of which is either working or failed. Consider an experiment that consists of
observing the status of each component, and let the outcome of the experiment be given by the vector (x1,x2,x3, x4, x5),
where xi is equal to 1 if component i is working and is equal to 0 if component i is failed.
(a) How many outcomes are in the sample space of this experiment?
(b) Suppose that the system will work if components 1 and 2 are both working, or if components 3 and 4 are both
working, or if components 1, 3, and 5 are all working. Let W be the event that the system will work. Specify all the
outcomes in W.
(c) Let A be the event that components 4 and 5 are both failed. How many outcomes are contained in the event A?
(d) Write out all the outcomes in the event AW.

The concept applied here is the probability of success and failure n(s) + n(f) = 1. i.e where one is successful and the other failed or one is working and the other is not working.

In the question, 5 components are given and the outcome are given by the vector (x1,x2,x3, x4, x5). where xi = 1 (component working) and xi = 0 (component failed).

To answer the first question of how many outcomes are in the sample space?

Each of the component x1 -x5 can either be of two possibilities (working or failed) i.e in two succession. recalling that the power of set is dependent on the number of elements in the experiment (n).

power = 2n (2 raised to power of n), where n = 5 (number of outcome of experiment)

hence 2 raised to power of 5 = 32

therefore, number of outcomes is 32

b)To answer the second question, if component 1 and 2 are working or if component 3 and 4 are working or if component 1, 3 and 5 are all working. it implies that;

i) component 1 and 2 working = (1,1,x3,x4,x5)

ii) component 3 and 4 working = (x1,x2,1,1,x5)

iii) component 1, 3 and 5 are all working = (1,x2,1,x4,1)

therefore the outcomes of W = { (1,1,x3,x4,x5), (x1,x2,1,1,x5), (1,x2,1,x4,1)}

c) To answer the third part, where component 4 and 5 are both failed. this implies the following;

i) only component 1 is working = (1,0,0,0,0)

ii) component 1 and 2 are working = (1,1,0,0,0)

iii) component 1 and 3 are working = (1,0,1,0,0)

iv) only component 2 is working = (0,1,0,0,0)

v) component 2 and 3 are working = (0,1,1,0,0)

vi) only component 3 is working = (0,0,1,0,0)

vii) component 1, 2 and 3 are working = (1,1,1,0,0)

viii) none of the component are working = (0,0,0,0,0)

in total, there are 8 outcomes. {(1,0,0,0,0),(1,1,0,0,0)

,(1,0,1,0,0), (0,1,0,0,0),(0,1,1,0,0),(0,0,1,0,0)

,(1,1,1,0,0),(0,0,0,0,0)}

d) to answer the last part

i) for event A, 4 and 5 are both failed = (1,1,1,0,0)

ii) for event W where 1 and 2 are working, 3 and 4 are working, 1 , 3 and 5 are working

={(1,1,1,0,0), (1,1,x3,x4,x5), (x1,x2,1,1,x5), (1,x2,1,x4,1)}