# Science:Math Exam Resources/Courses/MATH103/April 2005/Question 06 (b)

MATH103 April 2005

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### Question 06 (b)

A CT scanner in a hospital is switched on every morning and switched off at night. The scanner has a probability ${\displaystyle \displaystyle q=0.001}$ of failure when the technician turns it on in the morning. (Assume that if it turns on, it will not fail later that day i.e. treat this as a discrete event. Also assume that failure is independent of age or history of the machine.)

(b) Use your result in (a) to find the mean time of failure of this machine, i.e. the expected value of the number of the day on which it first fails. You may find it helpful to use the summation formula

{\displaystyle {\begin{aligned}\sum _{k=1}^{\infty }kr^{k-1}={\frac {1}{(1-r)^{2}}}\end{aligned}}}

for ${\displaystyle \displaystyle |r|<1}$

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