7. The follo wing diagram is a segment of a printed circuit which has eight rela

Question

7. The follo wing diagram is a segment of a printed circuit which has eight relays, used in a computer mother-board. Current will low from point "a" to point "b" if there is at least one closed path when the relays are switched to "close". However, the relays may malfunction Suppose each relay closes properly only with probability .80 when the switch is thrown, and suppose they operate independendy of one another (8) * b a. Let A, denote the event that the ith switch closes properly when the switches are thrown (i-1, . 8) and B denote the event that current can flow from point a to point b. Express event B in terms of event A's and then find the probability that current can flow from point a to point b. That is, find P(B). If the system is working, find the probability that the component 4 is working That is, find P ( A, B Are events B and A, independent ? Check! c.

Explanation / Answer

(a)

P(B) = P(current is flowing from point a to b)   

= P(A1) and P(A2) and (P(A3) or P(A4) ) and P(A5) and (P(A6) or P(A7)) and P(A8)

= 0.8 * 0.8 * (0.8 + 0.8 - (0.8*0.8)) * 0.8 * (0.8 +0.8-(0.8*0.8)) * 0.8

= 0.4096* 0.96*0.96 = 0.377

(b)

P( A4 | B) = P( B | A4 ) * P(A4) / P(B)

P(B) = 0.377

P(A4) = 0.8

P(B | A4) = P(A1) and P(A2) and P(A4) and P(A5) and (P(A6) or P(A7)) and P(A8)

                = 0.8 * 0.8 *0.8 * 0.8 * (0.8 +0.8-(0.8*0.8)) * 0.8

                = 0.314

P( A4 | B) =0.314*0.8 / 0.377 = 0.666

(c) TO check if events are independent:

Check if P( B ) and P( A4) = P(B) * P(A4)

If equal, events are independent , else not.

P(B) = 0.377      P(A4) =0.8

P(B) *P(A4) = 0.377*0.8 =0.3016

P(B and A4) = 0.666

They are not equal , hence events are not independent.

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