Please show equations and math used. The computing system at a large university

Question

Please show equations and math used.

The computing system at a large university is currently undergoing shutdown for repairs. Previous shutdowns have been due to hardware failure, software failure or power (electronic) failure. The system is forced to shutdown 73% of the time when it experiences hardware problems, 12% of the time when it experiences software problem, and 88% of the time when it experiences electronic problem. Maintenance engineers have determined that probabilities of hardware, software, and power problems are .01, .05, and .02 respectively. What is the probability that the current shutdown is due to hardware failure? Software problem? Power failure?

Explanation / Answer

here let probabilty of hardware faiure =P(H) =0.01 ; software =P(S) =0.05 ; power failure =P(P) =0.02

also probability of forced shut down =P(F)

here probabilty of forced shutdown given hardware failure =P(F|H) =0.73

here probabilty of forced shutdown given software failure =P(F|S) =0.12

here probabilty of forced shutdown given power failure =P(F|P) =0.88

therefore probability that current shutdown is due to hardware failure =P(FnH) =P(H)*P(F|H) =0.01*0.73=0.0073

therefore probability that current shutdown is due to software failure =P(FnS) =P(S)*P(F|H) =0.05*0.12=0.006

therefore probability that current shutdown is due to power failure =P(FnP) =P(P)*P(F|P) =0.02*0.88=0.0176

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