A factory has two assembly lines, each of which is shut down (S), at partial cap

Question

A factory has two assembly lines, each of which is shut down (S), at partial capcity (P), or at full capacity (F).

Probability values for assembly line operations:

(S,S)=.02     (S,P)=.06     (S,F)=.05

(P,S)=.07     (P,P)=.14     (P,F)=.20

(F,S)=.06     (F,P)=.21     (F,F)=.19

Evaluate the following scenarios:

(a) Both lines are at full capacity conditional on neither line being shut down.

(b) At least one line is at full capacity conditional on neither line being shut down.

(c) One line is at full capacity conditional on exactly one line being shut down.

(d) Neither line is at full capacity conditional on at least one line operating at partial capacity.

Explanation / Answer

(a) neither line being shut down, so ignore all above pairs with S in it
left with (P,P), (P,F), (F,P) (F,F)

Probability = (F,F) /[(P,P)+(P,F)+ (F,P) +(F,F)] = 0.19/(0.14+0.2+0.21+0.19) = 0.26

(b) Probability = (P,F) + (F,P) + (F,F) /((P,P)+ (P,F) + (F,P) + (F,F)) = (0.2+0.21+0.19)/(0.14+0.20+0.21+0.19) = 0.81

(c) look for all pairs with exact one S in it:
(S,P) (S,F) (P,S) (F,S)

Probability = (S,F) + (F,S) / [(S,P) + (S,F) + (P,S) + (F,S)] = (0.05+0.06)/(0.06+0.05+0.07+0.06) = 0.46

(d) look for all pairs with at least one P in it

(S,P) (P,S) (P,P) (P,F) (F,P)

Probability = (S,P)+ (P,S) +(P,P) /[(S,P) + (P,S)+ (P,P)+ (P,F)+(F,P)] = (0.06+0.07+0.14)/(0.06+0.07+0.14+0.20+0.21)

= 0.40

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