Description:

Responsible companies have regular inspection and maintenance procedures for the engines on their fleet of planes. During such inspections, the fingers of the blades are inspected for SCC cracks. Consider a company whose aero engines use blades with 8 fingers and each disc within the engine has 80 blades attached to it. These blades are designed with safety in mind and so blades will only break away (i.e. fail) from a rotating disc during flight if 3 or more fingers have such a crack defect. In turn it only requires one blade to fail for the disc to fail. Such events would of course be catastrophic during flight.

During a routine inspection of a blade from a disc within one of its engines, a finger was selected at random and an SCC crack was observed in this finger. The critical question is then do any of the other fingers in this blade/disc unit have such cracks and if so does this warrant the grounding of all its planes using this engine design.

To address this issue, the engineers are instructed to randomly select an additional 79 fingers within this disc/blade unit and test them for SCC cracks, whilst the fleet remains fully operational. (This testing is very expensive, but not as expensive as grounding all the planes). This was done and no further cracks were found. On this result senior management concluded their planes were safe for flight and so do not order a general grounding of their fleet.

The critical question from an engineering and legal perspective is, should the fleet have been grounded. Assume the number of failed fingers on a blade and the number of failed blades on a disc follow a binomial distribution when answering the following questions to come to a conclusion about whether this was the right decision.

Instruction:

PROVIDE ALL YOUR NUMERICAL ANSWERS TO 4 DECIMAL PLACES!

Question 1

A random variable that has a binomial distribution has a varying probability of success?

o True
o ?False

Question 2

Consider the following experiment that was used to generate an observation on a random variable. A rotating drum is filled with many plastic tags that have either a zero or a one written on them. The drum is rotated, a tag removed and the number written on it is recorded. The tag is then discarded and the above described process is repeated a further n = 5 times. The random variable of interest in this experiment is the sum of the numbers shown on these 5 tags. What distribution does this variable have?
o Poisson.
o Binomial
o Hypergeometric.
o Normal.

Question 3
Based on the information available after the additional testing, what is the probability of an SCC crack existing in a finger?

Question 4
What is the mean number of failed fingers in a single blade?

Question 5
What is the standard deviation in the number of failed fingers in a single blade?

Question 6
When testing all 8 fingers on a single blade, how many ways are there of observing 4 fingers with an SCC crack?

Question 7
What is the probability of exactly 1 finger failing on a single blade?

Question 8

What is the probability of a blade failing?

Question 9
What is the probability of a disc not failing? (Use an unrounded value for the probability of a blade failing.

Question 10
From a safety of life point of view and based on your last answer, do you think management were correct in their decision to allow the fleet to fly without grounding more planes and testing their disc/blade units?
o Yes
o No

Question 11
Suppose upon the testing of all blades on the blade/disc unit no further SCC cracks were found, what is the revised probability of a disc not failing? Express your answer in PERCENT (to 4 decimal places) but do not round any values during your calculations.