5.5.1.1 Bayes' Example (Discrete
Distribution)
To demonstrate the use of Bayes' Equation within the framework
of the binomial estimation of reliability, consider the following simplistic
(but illustrative) example.
We wish to estimate the reliability of a simple pyrotechnic
device which, upon being tested, either fires (success) or doesn't fire
(failure). We have in the warehouse two lots of this component, one of which
we have been assured has a reliability of R = 0.9 (that is, in the long term,
9 of 10 randomly selected components will work). The other lot supposedly
contains only 50% good items. Unfortunately, we have lost the identity of
which lot is which.
After randomly selecting one of the lots (such that the
probability for each lot is 0.50), we then randomly select a single item from
it (each item has equal chance of being chosen), which fails in test. What can
be said about all this in the context of Bayesian analysis?
First, terms must be defined (see Figure 5.51).
A_{1} 

“Lot chosen has R =
0.50” 



A_{2} 

“Lot chosen has R =
0.90” 
Then, from above,
Pr [A1] =
0.5, Pr [A2] =
0.5.