Sheldon Ross 10 : Example 2.8

This example has already been done neatly in the textbook, but this is an attempt to explain the ideas visually.

Question: “Suppose that an airplane engine will fail, when in flight, with probability 1−p independently from engine to engine; suppose that the airplane will make a successful flight if at least 50 percent of its engines remain operative. For what values of p is a four-engine plane preferable to a two-engine plane?”


Part 1: Visualization

Consider the check and cross marks to be the engines. A check indicates a functional engine and a cross indicates a failed engine.

Success configurations for a 2-Engine Plane

 

Success configurations for a 4-Engine Plane

 

 

Success configurations for a 6-Engine Plane

 

Success configurations for a 8-Engine Plan

 

The code is general enough to be extended to 2n engines. Try it yourself!


Part 2: Mathematical Modelling

Mathematical modelling is straightforward since the objective is clear. For a plane with 2n engines, we need at-least n engines working. Total probability for a successful flight is

∑ nCk pk(1-p)2n-k , where k ≥ n

The question asks the probability for which the 2-Engine Plane is preferable to the 4-Engine Plane. For this we need 2n be equal to 2 and 4. The expansion results in the following.

2p(1 − p) + p2 = 6p2 + (1 − p)2 + 4p3(1 − p) + p4

Solving for p, we get the following solutions. p = 0; p = 0.66667; p = 1. The code for solving that is pasted below.

Graphically, and comparing the 2-Engine Plane to the 4-Engine planes, we see a nice pattern taking shape.

General behavior of the 2n-Engine probabilities

The Mathematica notebook that was used for the analysis is attached as a PDF here. Please let me know if you have any questions.


End of the post


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