A water network has three connections, C1,C2,C3, as shown in Figure 1.18. For each connection, at the places marked 1, 2, 3, some switches have been put and at a particular instant, any switch can be either ON (thus allowing water to pass through) or OFF. A connection is considered to be working if water can pass from point A to point B.

(i) Give a suitable sample space to describe the possible switch positions in the network.

(ii) For each connection, identify the events that describe the working status of the connection.

(iii) Let us denote by pi, i = 1, 2, 3, the probability that the switch i is in the ON position and by pij for i ≠ j the probability that both switches i and j are ON

(i = 1, 2, 3 and j = 1, 2, 3). Finally, p123 denotes the probability that all three switches are in the ON position. Express the probabilities of the events found in Part (ii) in terms of p1, p2, p3, p12, p13, p23, and p123.