You had a wonderful Time Machine and you would take it out for a spin every week

Question

You had a wonderful Time Machine and you would take it out for a spin every weekend. Unfortunately it stopped working. You know that in 8% of cases the cause is a fault flux capacitor. Dr. Brown has a test diagnostics equipment. If the Flux Capacitor is broken the test will show it in 95% of the time. At the same time, if the Flux Capacitor is working, the test will mistakenly identify it as broken in 10% cases. You run the test and it shows that the Flux Capacitor is broken. What is the probability that it is actually broken? Express your answer as a percentage; for example, the answer 3.18% should be entered as 3.18 without the percentage sign.

Explanation / Answer

Using Baye's Theorem
P(Broken|Test shows broken)=P(Test shows broken|Broken)*P(Broken)/(P(Test shows broken|Broken)*P(Broken)+P(Test shows broken|Not Broken)*P(Not Broken))=0.95*0.08/(0.95*0.08+0.1*0.92)=0.452380952

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