Unizor - Probability - Advanced Problems 5

Problem A
Imagine you are a member of a tourist group in Bangkok, Thailand in front of the Emerald Buddha Temple. There are N people in your group and all of them have the same kind of shoes of the same size. To enter, all of you have to take off your shoes. You do, enter the temple and, when you got out, each of you looks for the shoes at the entrance and sees that they are all the same (left and right shoes differ, but all the left and all the right are the same).
What is the probability that each of you will put on his or her own pair of shoes?

Answer: (1/N!)^2

Problem B
A city is supplied with electricity from the power station through two substations connected sequentially. The probability of an accident on each substation during a period of one year when the electricity is interrupted is P.
(a) What is the probability of uninterrupted flow of electricity to a city during a period of one year?
(b) What should be the value of P if you would like the probability of uninterrupted supply of electricity during a period of one year to be 0.99?
(c) What is the probability of uninterrupted flow of electricity to a city during a period of one year if it's sequentially connected through N substations?

Answer:
(a) (1−P)^2
(b) P = 1−√0.99 ≅ 0.0050
(c) (1−P)^N

Problem C
The same problem as B above, but the substations are connected in parallel.

Answer:
(a) 1−P^2
(b) P = √(1−0.99) = 0.1
(c) 1−P^N

Problem D
A city needs a solution to some specific engineering problem. There are N companies specializing in this field, each with its own rate of success.
Let's set the probability of a company #i to be able to solve the problem during a specified time to be Pi.
The city executive invites these companies in sequence, from company #1 to company #N.
If a particular company cannot solve this problem in a specified time, the city executive asks the next company to try. If all N companies failed to provide a solution, each is given another period of time to think about it, also in sequence. This process continues until a solution is found or until each company tried and failed T times. In case of the latter, the city gives up on a project as unsolvable.
What is the probability that the company #K will be the one that solves the problem?

Answer:
ProbK = Σj∈[1,T](A^(j−1)·BK)
where
A = Πn∈[1,N](1−Pn)
BK = PK·Πn∈[1,K−1](1−Pn)