Question

1. Assume that a component passes a test is 0.85 and that components perform independently. What is the probability that the third failure will occur on the tenth component tested?
2. Consider the distribution of problem 1. Graph this distribution.

Answer 1

The distribution which tells the probability of rth success on xth trial is the negative binomial distribution.

P(X = x) = P(r-1 successes in x-1 trials) * P(success on xth trial)

P(X = x) = P(X = r-1) like in binomial distribution with x-1 trials * P(success on xth trial)

=> =

Here, x = 10, r = 3 with p = 1-0.85 = 0.15 [ because P(failure) is asked ]

=> = 0.039

The graph will look something like this (not to scale):