Before the distribution of certain statistical software, every fourth CD is test

Question

Before the distribution of certain statistical software, every fourth CD is tested for accuracy. The testing process consists of running four independent programs and checking the results. The failure rates for the four testing programs are. respectively, .01, .03, .02, and .01. What is the probability that a CD was tested and failed any tests? Given that a CD was tested, what is the probability that it failed program 2 or 3? In a sample of 100, how many CDs would you expect to be rejected? Given that a CD was defective, what is the probability that it was tested?

Explanation / Answer

Prob{Checked} = 0.25

Prob{Fail #1} = 0.01
---->   Prob{NOT Fail #1} = 1.0- (0.01) = 0.99

Prob{Fail #2} = 0.03
---->   Prob{NOT Fail #2} = 1.0- (0.03) = 0.97

Prob{Fail #3} = 0.02
---->   Prob{NOT Fail #3} = 1.0- (0.02) = 0.98

Prob{Fail #4} = 0.01
---->   Prob{NOT Fail #4} = 1.0- (0.01) = 0.99

(a): Prob{CheckedANDFailed AnyCheck} =
          = Prob{Checked}*Prob{NOT (Pass All Checks)}
          = (0.25)*( 1 - (0.99)*(0.97)*(0.98)*(0.99) )
          = 0.017079

(b): Prob{Failed #2 OR Failed#3 GIVENChecked}=
          = Prob{NOT Pass Both#2 & #3}
         = 1 - (0.97)*(0.98)
          = 0.0494

(c): {Expected Rejections Out Of100} =
          = (100)*Prob{NOT(Pass All Checks)}
          = (100)*(1 - (0.99)*(0.97)*(0.98)*(0.99))
         = 6.8317

(d): Prob{CheckedGIVENDefective} =
          = Prob{Checked}    (<--- checks are independent ofquality)
          = 0.25

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