Some auto accidents are due to environmental factors, 50 % of these accidents ar

Question

Some auto accidents are due to environmental factors, 50 % of these accidents are due to slick roads, 17 % are due to glare, 11 % are due to visual obstructions, and 22 % are due to other environmental reasons. (Source: National Highway Traffic Safety Administration) Once it has been determined that an accident has been caused by environmental factors, a group of investigators tries to determine if slick roads caused the accident. This group of investigators has the following record: 88 % of the time that slick roads was the cause of the accident, the investigators concluded that slick roads was the cause. 13 % of the time that glare was the cause of the accident, the investigators concluded that slick roads was the cause. 7% of the time that visual obstructions was the cause of the accident, the investigators concluded that slick roads was the cause. 18% of the time that other factors was the cause of the accident, the investigators concluded that slick roads was the cause. What is the probability that an accident was caused by glare given that the investigators concluded that the accident was caused by slick roads?

Explanation / Answer

let probabilty of accident by slick road =P(S), by glare =P(G), by visual =P(V) and by other factors =P(O)

hence probabilty that investigators concluded accident by slick roads=P(ICS)

=P(S)*P(ICS|P(S)+P(G)*P(ICS|P(G)+P(V)*P(ICS|P(V)+P(O)*P(ICS|P(O)

=0.5*0.88+0.17*0.13+0.11*0.07+0.22*0.18 =0.5094

hence probabilty that an accident was caused by glare given that the investigators concluded that the accident was caused by slick roads =P(G|ICS) =P(G)*P(ICS|P(S)/P(ICS) =0.17*0.13/0.5094 =0.0434

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