The “Coupon Collector's Problem” and Quality Control

ABSTRACT

This article examines the classical “coupon collector's problem” and applies several key results to certain problems in quality control sampling. It is shown how solutions to the coupon sampling problem are readily adaptable to industrial sampling problems where the intent is to sample at least 1 of k specific product unit types in a well mixed stream of product where the product unit is equally likely to be any of the k specific types. Several variations on this theme are presented. Theoretical formulas are developed and applied to quality control sampling. Additional applications involving Monte Carlo simulation using the freeware programming language “R” are also illustrated.

KEYWORDS:

• coupon collector's problem
• probability
• quality control
• sampling

ACKNOWLEDGEMENTS

Special thanks to a friend and colleague, Mr. Dave Arnold, for his initial inquirery regarding this problem as applied to the injection molding industry. The author is also grateful to Pratt and Whitney for their kind permission to publish this paper and to the referee for helpful suggestions and comments during the development of this paper.

Notes

¹In this section N is used to denote the random variable defined as sample size, and n is used to denote a specific value of N.

²The random variable x has a geometric distribution when x is defined as the number of independent trials required to produce the first “event” where the event probability, p, remains constant from trial to trial. The expected number of such trials is 1/p.

³All tables generated using MS Excel.

⁴This formula is an algebraic variant of the Johnson and Kotz (1969) formula.

Note: “Freq” = Frequency; “CumeFreq” = Cumulative Frequency; “RelFreq” = RelativeFrequency “CRF” = Cumulative Relative Frequency.

Note: “Freq” = Frequency; “CumeFreq” = Cumulative Frequency; “RelFreq” = Relative Frequency “CRF” = Cumulative Relative Frequency.

⁵This follows from the independence of the selection process from sample to sample. Note also that this argument can be extended to two or more missing units in the sample. For example, for any two distinct units missing, the probability is (1 − 2/k) ⁿ .