13. Responsible companies have regular inspection and maintenance procedures for the

engines on their fleet of planes. During such inspections, the fingers on the turbine blades

within engines are inspected for SCC cracks. Consider a company whose aero engines use

blades with 6 fingers and each disc within the engine has 50 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 2 or more fingers have such crack defects. In turn it only

requires one of the blades to fail to cause the disc to fail. (i.e. 1 or more). 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 concern now is that such cracks may be present in all its other aircraft

engines (which if true would warrant the grounding of all its planes using this engine

design). To address this issue, the engineers are instructed to randomly select an additional

49 fingers within its engine fleet and test them for SCC cracks - whilst the fleet remains

fully operational. This was done and no further cracks were found. Expressing probability

as a number between 0 and 1, answer the following questions to 4 decimal places (and

never specify units or enter text of any kind): Based on the information available after the

additional testing, the probability of an SCC crack existing in a finger is

The mean number of failed fingers in a single blade is

When testing all 6 fingers on a single blade, the number of ways of observing 3 fingers with

an SCC crack is

The probability of exactly 3 fingers failing on a single blade is

The probability of a blade failing is

The probability of a disc not failing is