Probability of Reliability.nb

Probability of Reliability

Robert Harder
robert.harder@afotec.af.mil
October 2003
Download the source Mathematica notebook.
View a MathML-enabled version of this document.

Introduction

The question before me is this: "Given n tests of a system and y failures, what is the probability that the system's reliability is r or better?
I have found in all my statistical text books and on several web pages discussions of finding the probability of failure given a system's reliability, but I have not discovered any writing that goes the other direction: finding the system's reliability. In my field of Operational Test and Evaluation (OT&E) we often need to estimate a system's reliability.
Note #1: Probability equations generally refer to the variable p as the proportion of "successes." In our terminology a success is a system failure (how do you like that?). In other words, system reliability r is equal to (1-p).
Note #2: This document was constructed as a Mathematica notebook. I hope that even if you are not familiar with Mathematica notations, you will still be able to follow the important points.

Plotting Failure Probability

The function ThreeWayPDF is a construct for embedding three versions of the following familiar expression with the variable y taking on values of zero, one, and two.

In[1]:=

$ThreeWayPDF = Table[Binomial[n, x] p^x (1 - p)^(n - x), {x, 0, 2}]/.p-> (1 - r)$

Out[1]=

${r^n, n (1 - r) r^(-1 + n), 1/2 (-1 + n) n (1 - r)^2 r^(-2 + n)}$

The following plot shows the reliability of a system in 28 tests (n→28). The vertical axis is the probability of y failures occurring in 28 tests if the system reliability is r (horizontal axis).
Examples of reading the table:
"If a system's reliability is 95%, there is roughly a 25% chance of getting zero failures in 28 tests."
"If the system's reliability is 90%, there is roughly a 20% chance of getting exactly one failure in 28 tests."

In[12]:=

RowBox[{RowBox[{P13, =, RowBox[{Plot, [, RowBox[{Evaluate[ThreeWayPDF/.n28], ,, RowBox ... aphics[Text["y=1", {.955, .4}]], Graphics[Text["y=2", {.90, .28}]]] ;

[Graphics:HTMLFiles/index_5.gif]

Plotting Reliability Probability

Look at the next plot and see how I have highlighted the areas around 95% reliability for y=1 failure. If the y=1 curve shows all combinations of reliability and probability, then the darker portion under the curve from 95% to 100% reliability should represent the probability that the reliability is greater than 95%.

In[14]:=

In[15]:=

$RowBox[{RowBox[{P1, =, RowBox[{FilledPlot, [, , , RowBox[{n (1 - r) r^(-1 + n)/.n-> ... br /> Reliability vs. Probability of Failure, n=28 tests"}], ]}], ;}]$

[Graphics:HTMLFiles/index_8.gif]

Area Under the Curve

To find the probability that the reliability is r or greater, we need to calculate what proportion of the area under the curve is represented, in our example above, by the darker portion. To do this we divide the area of the darker portion by the total area under the curve. Note the limits of integration in the ProbOfReliability formula:

In[18]:=

ProbOfReliability[r_, n_, y_] := (∫_0^(1 - r) (n) p^y (1 - p)^(n - y) p)/(∫ ... y y

This function can be plotted to give a continuous representation of the probability of any reliability level. Plotting multiple values of y on a single chart gives a good picture of what statements could be made regarding the reliability of a system after n tests.

In[22]:=

$SDBPatch = Import["C:\\temp\\Patch.jpg"] ; PY[yy_] := Plot[Evaluate[ProbOfReliabilit ... otStyle {Hue[1 - yy/5], Thickness[.004]}, PlotPoints50] ; P05 = Array[PY, 6, 0] ;$

In[25]:=

RowBox[{RowBox[{SDBPlot, =, RowBox[{Show, [, RowBox[{Graphics[Rectangle[{.75, .01}, {.80, .19} ... wBox[{RowBox[{GrayLevel, [, 0.9, ]}], ,, AbsoluteThickness[1]}], }}]}], }}]}], }}]}]}], ]}]}], ;}]

Final Plot

In[26]:=

[Graphics:HTMLFiles/index_13.gif]

Some ways to read the final plot:
If zero failures occurred, there is a 77% chance ("confidence") that the system reliability is 95% or better.
If one failure occurred, there is a 43% chance that the system reliability is 95% or better.
If one failure occurred, there is an 80% chance that the system reliability is 90% or better.

Created by Mathematica (October 14, 2003)