question relating to picture: a) what is the probability, on any day, that it is

Question

question relating to picture:
a) what is the probability, on any day, that it is possible to drive from point A to point B?

b) Given that it is possible to drive from point A to point B, what is the probability that road 3 is open that day?

3. Consider the configuration of roads as shown below: there are 5 roads labeled R1 through R5 each given by a straight line segment. Suppose that because of road construction, on any given day, each road is only open with probability 1/3, and that all the roads are open independently of the others. R1 R4 R3 RS R2

Explanation / Answer

a)P(possible to drive from A to B) =P(R1R4 U R2R5 U R1R3R5 U R2R3R4)

=P(R1R4)+P(R2R5)+P(R1R3R5)+P(R2R3R4)-P(R1R2R4R5)-P(R1R3R4R5)-P(R1R2R3R4)-P(R1R2R3R5)-P(R2R3R4R5)-P(R1R2R3R4R5)+P(R1R2R3R4R5)+P(R1R2R3R4R5)+P(R1R2R3R4R5)+P(R1R2R3R4R5)-P(R1R2R3R4R5)

=(1/3)*(1/3)+(1/3)*(1/3)+(1/3)*(1/3)*(1/3)+(1/3)*(1/3)*(1/3)-(1/3)*(1/3)*(1/3)*(1/3)-(1/3)*(1/3)*(1/3)*(1/3)-(1/3)*(1/3)*(1/3)*(1/3)-(1/3)*(1/3)*(1/3)*(1/3)-(1/3)*(1/3)*(1/3)*(1/3)-(1/3)*(1/3)*(1/3)*(1/3)*(1/3)+(1/3)*(1/3)*(1/3)*(1/3)*(1/3)+(1/3)*(1/3)*(1/3)*(1/3)*(1/3)+(1/3)*(1/3)*(1/3)*(1/3)*(1/3)+(1/3)*(1/3)*(1/3)*(1/3)*(1/3)-(1/3)*(1/3)*(1/3)*(1/3)*(1/3) =59/243 =0.242798

b)

P(A -B) =59/243

P(A-B and road 3 is block) =P(R1 R4 or R2R5)=P(R1R4)+P(R2R5)-P(R1R2R4R5)

=(1/3)*(1/3)+(1/3)*(1/3)-(1/3)*(1/3)*(1/3)*(1/3) =17/81

hence P(road 3 open|A-B)=1-P(road 3 block|A-B) =1-P(A-B and road 3 is block)/P(A-B)

=1-(17/81)/(59/243)=1-51/59 =8/59

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