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.

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.

Fearen · Accepted Answer

Step 1 Let X be the number of engines in working condition Given information: Number of engines (n)=4 Probability of an engine to work (p)=0.92 X follows Binomial Distribution Xsin⁡B∈(4,0.92) The P.M.F for binomial distribution is given by P(X=x)=(nx)×px×(1−p)n−xP(X=x)=(nx)×px×(1−p)n−x Step 2 a) Probability that an aircraft in flight can land safely From the given information An aircraft can land safely only when at least 2 engines are in working condition P(X≥2)=1−P(X&amp;lt;2)=1−(P(X=0)+P(X=1)) =1−((40)×0.920×(1−0.92)4−0+(41)×0.921×(1−0.92)4−1)=1−((40)×0.920×(1−0.92)4−0+(41)×0.921×(1−0.92)4−1) =1−0.00004096−0.001882416=0.99807 Probability that an aircraft in flight can land safely is 0.99807 Step 3 b) Minimum value of p for which probability of landing safely must be at least 99.5%. P(X≥2)≥0.995 Using Binomial distribution table Taking p=0.90 P(X≥2)≥0.9963 Taking p=0.89 P(X≥2)≥0.9951 For getting the exact value, we can use linear interpolation 0.90−(0.90−0.89)(0.9963−0.9951)×(0.9963−0.995)=0.889 So, the value of p is 0.889 P(X&amp;lt;2)&amp;lt;0.001 From Binomial Distribution Table For p=0.90 P(X&amp;lt;2)=0.0037 When p=0.95 P(X&amp;lt;2)=0.0005 For getting the exaxt value, we can use linear interpolation 0.90+(0.95−0.90)(0.0037−0.0005)×(0.0037−0.001)=0.9422 So, the value of p is 0.9422 Step 4 c) Minimum number of engines that would achieve at least 99.5% probability of landing safely P(X≥2)≥0.995 1−P(X&amp;lt;2)≥0.995 1−(n0)×0.920×(1−0.92)n−0−(n1)×0.921×(1−0.92)n−1

A commercial jet aircraft has four engines. For an aircraft in flight

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