Question

Failures of a testing instrument from contamination particles on the product follow a Poisson process with a mean of 0.32 failures per an 8-hour shift.

a) Find the probability that the instrument does not fail during a shift.

b) Find the probability that there are more than 3 failures in a 48-hour period.

c) Provided there is at least one failure in a 24-hour period, what is the probability that there are no more than 2 failures during that period.

Thank you very much!

Answer 1

a)

probability that the instrument does not fail during a shift =P(X=0)=e^-0.32*0.32⁰/0!=0.726149

b)expected number of failures in 48 Hours =0.32*48/8=1.92

probability that there are more than 3 failures in a 48-hour period=P(X>3)

=1-P(X<=3)=1-(P(X=0)+P(X=1)+P(X=2)+P(X=3))

=1-(e^-1.921.92⁰/0!+e^-1.921.92¹/1!+e^-1.921.92²/2!+e^-1.921.92³/3!)=0.128737

c)

expected number of failures in 24 Hours =0.32*24/8=0.96

hence P(at least one failure)=P(X>=1)=1-P(X=0)=1-e^-0.960.96⁰/0! =0.6171

P(no more then 2 failure and at least one failure)=P(X=1)+P(X=2)=e^-0.960.96¹/1!+e^-0.960.96²/2!

=0.5440

hence probability that there are no more than 2 failures given at least one failure

=0.5440/0.6171 =0.881556