confidence of a failure

Question

A joint of steel pipe (casing) has a 1% failure rate. 400 joints of casing are in a typical well. How may wells can I drill or joints can I run, before I am 90% confident of having at least one casing failure? You can leave out the well to simplify the problem: Casing has a 1% failure rate. How many joints of casing can I have before I am 90% confident in having a failure? Thanks for re teaching me. I think it is called gambler's ruin.

Answer 1

This is a problem using the geometric distribution. Distributions with that name are defined in different ways.

My definition is that $X$ is the number of the trial on which we see the first casing failure.

Then the PDF of $X$ is $f(k) = P(X = k) = (1-p)p^{k-1},$ for $k = 1, 2, \dots .$ Also, the CDF is $F(k) = P(X \le k) = \sum_{i=1}^k f(i).$

We can use R statistical software to make a table of the PDF and CDF for your problem and then print out the first few and the last few rows of the table:

 > x = 1:230;  pdf = .01*.99^(x-1);  cdf=cumsum(pdf)
 > TAB = cbind(x, pdf, cdf)
 > head(TAB); tail(TAB)
      x        pdf        cdf
 [1,] 1 0.01000000 0.01000000
 [2,] 2 0.00990000 0.01990000
 {3,] 3 0.00980100 0.02970100
 [4,] 4 0.00970299 0.03940399
 [5,] 5 0.00960596 0.04900995
 [6,] 6 0.00950990 0.05851985
        x         pdf       cdf
[225,] 225 0.001052649 0.8957877
[226,] 226 0.001042123 0.8968299
[227,] 227 0.001031701 0.8978616
[228,] 228 0.001021384 0.8988830
[229,] 229 0.001011170 0.8998941
[230,] 230 0.001001059 0.9008952

It seems that it will take 230 fittings to be 90% sure of a casing failure. I will leave it to you to figure out how to sum the geometric series to get the CDF. Then you can solve to find the smallest $k$ such that $F(k) \ge .9.$ (Of course, I had to do that in order to know how long to make the table.)

Note: The version of the geometric distribution programmed into R is for the random variable $Y$ which counts the trials up to, but not including, the first casing failure. Thus in R, qgeom(.9, .01) returns 229.