Question

A commercial jet aircraft has four engines. For an aircraft in flight to land
safely, at least two engines should be in working condition. Each engine has an independent
reliability of p 92%.
a. What is the probability that an aircraft in flight can land safely?
b. If the probability of landing safely must be at least 99.5%, what is the minimum
value for p? Repeat the question for probability of landing safely to
be 99.9%.
c. If the reliability cannot be improved beyond 92% but the number of
engines in a plane can be increased, what is the minimum number of
engines that would achieve at least 99.5% probability of landing safely?
Repeat for 99.9% probability.
d. One would certainly desire 99.9% probability of landing safely. Looking
at the answers to (b) and (c ), what would you say is a better approach
to safety, increasing the number of engines or increasing the reliability of
each engine?

Answer 1

Given that Each engine has an independent reliability of . The jet aircraft has four engines . Let .

The number of engines that works correctly out of has Binomial distribution.

The PMF of is  

For an aircraft in flight to land safely, at least two engines should be in working condition. In reliability terminology this is 2 out of 4 system.

a) The the probability that an aircraft in flight can land safely is

b) We need . So,

Use R to solve: minimum

Use R to solve: minimum

c) Here is fixed.

Minimum .

Minimum .

d) Increasing the reliability of each engine is better than increasing the number of engines . If you increase the reliability of each engine you will get more tha 99.5% safety.