Question

Q Computers has invented quantum computers. Each computer contains an exotic sub-atomic particle. Unfortunately this particle decays in the same
manner as all radioactive particles. Therefore an average quantum computer only lasts for 14 months.
The University has purchased one of these computers and Professor Squiggle wants to use it for 7 months. When he tries to book it he finds that it is
already booked out for the first 5 months. So he books it for the next 7 months. What is the probability that the computer will fail during the time that
professor Squiggle is using it (not before and not after)? (Answer to 2 decimal places)
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Professor Squiggle is so impressed with the quantum computer that he decides to apply for a grant to purchase his own for another experiment. However
the granting body will not give him a grant unless he has at least a 87% chance of successfully completing the experiment. What is the longest experiment
(in months) that Professor Squiggle could run and still expect to have this likelihood of the computer not failing over the time? Answer to 2 decimal places.
____________

Answer 1

The age of average quantum computer = 14 months

so here the probability of x, where x is the life of a random quantum computer.

f(x) = (1/14) e^-x/14

of the cumulative probabilty distribution is

F(x) = 1 - e^-x/14

Pr(5 months < x < 12 months) = F(12) - F(5) = [1 - e^-12/14] - [1 - e^-5/14] = 0.5756 - 0.3003 = 0.2753

(ii) Here it should be 87% chance of successfully completing the experiment.

so as we know here that

Pr(x < x₀) = 0.87

where x₀ is the maximum run the quantum computer had.

Pr(x < x₀) = 0.87

1 - e^-x₀/14 = 0.87

e^-x₀/14 = 0.13

x₀/14 = 2.04022

x₀ = 28.56months