Question # 8 There are 3 types of replaceable parts, A, B and C ina circuit-boar

Question

Question # 8 There are 3 types of replaceable parts, A, B and C ina circuit-board unit. The fractions of A, B and C in the circuit are: 20 %, 30 % and 50% respectively. Further, the chance that Part A can survive in excess of 100 hours of operation is 70 %. Likewise, the probabilities that Part B and C can survive in excess of 100 hours respectively are 0.4 and 0.3. (a) Find the probability that a randomly observed part on the board is likely to survive 100 hours of operation; and, (b) suppose a part is observed to have survived 100 hours of operation, determine the conditional probability that this part belongs to the type j, where j 1,2 or 3. Answer hints: (a): 41%) Question # 9 There are 3 space-mission sites, X, Y and Z operated proportionally to the extents of 35 %, 30 % and 35% respectively. Further, unsuccessful missions observed fronm these sites are in the proportion 1:2:3 respectively. Further, the chance that Part A can survive in excess of 100 hours of operation is 70 %. Suppose a certain mission, M commissioned by these sites is declared as being unsuccessful, what is the probability that M has originated from: (i) X; or from (ii) Y. Answer hints: (i): 0.175 and) 0.300;

Explanation / Answer

Ans:

8)Given that

P(A)=0.2

P(B)=0.3

P(C)=0.5

P(>100 hr/A)=0.7

P(>100 hr/B)=0.4

P(>100 hr/C)=0.3

a)P(>100 hrs)=0.2*0.7+0.3*0.4+0.5*0.3=0.14+0.12+0.15=0.41

b)Use bayes' rule

P(A/>100 hr)=P(>100 hr/A)*P(A)/P(>100 hr)=0.7*0.2/0.41=0.14/0.41=0.3415

P(B/>100 hr)=P(>100 hr/B)*P(B)/P(>100 hr)=0.4*0.3/0.41=0.12/0.41=0.2927

P(C/>100 hr)=P(>100 hr/C)*P(C)/P(>100 hr)=0.3*0.5/0.41=0.15/0.41=0.3659

b)

P(X)=0.35

P(Y)=0.30

P(Z)=0.35

P(unsuccesful/X)=1/6

P(unsuccesful/Y)=2/6

P(unsuccesful/Z)=3/6

P(unsuccessful)=0.35*(1/6)+0.3*(2/6)+0.35*(3/6)=0.333=1/3

i)P(X/unsuccesful)=P(unsuccessful/X)*P(X)/P(unsuccessful)=[(1/6)*0.35]/(1/3)=0.35*0.5=0.175

ii)

P(Y/unsuccesful)=P(unsuccessful/Y)*P(Y)/P(unsuccessful)=[(2/6)*0.30]/(1/3)=0.300

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