Introduction to Probability · Problem 5.1.35

Distributions and Densities — Grinstead & Snell

Mathematics — Probability · hypergeometric-distribution

Problem

A manufactured lot of brass turnbuckles has

S

items of which

D

are defective. A sample of

s

items is drawn without replacement. Let

X

be a random variable that gives the number of defective items in the sample. Let

p(d) = P(X = d)

. (a) Show that

p(d) = \frac{\binom{D}{d}\binom{S-D}{s-d}}{\binom{S}{s}}.

Thus,

X

is hypergeometric. (b) Prove the following identity, known as Euler's formula:

\sum_{d=0}^{\min(D,s)} \binom{D}{d}\binom{S-D}{s-d} = \binom{S}{s}.