. Suppose that a certified public accountant (CPA) has found that 15% of the com

Question

.   Suppose that a certified public accountant (CPA) has found that 15% of the company audits contain substantial errors. If the CPA audits a series of accounts (assuming each audit is independent from the next), find the following:

(a) What is the probability that the first account containing substantial errors is the fourth one to be audited?

(b) What is the probability that the first account containing substantial errors will occur on or after the third audited account?

(c) What’s the expected number of audits until one with substantial errors is found?

Explanation / Answer

The number of audits at which the first account containing substantial errors will occur (X) follows a Negative Binomial Distribution with k = 1 and p = 0.15

P(X = n) = (1-p)n-1 p

a. P(X = 4) = (1-p)3 p = (1-0.15)3 0.15 = 0.0921

[ X = 4 means the first 3 didnt have errors andthe 4th one had error ]

b. P(X >= 3) = 1 - P(X=1) - P(X=2)

= 1 - p - (1-p) p

= 1 - 0.15 - 0.85*0.15 = 0.7225

c. E(X) = k / p = 1 / 0.15 = 6.6667

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