Mathematically, a sampling design is denoted by the function ${\displaystyle P(S)}$  which gives the probability of drawing a sample ${\displaystyle S.}$ 

During Bernoulli sampling, ${\displaystyle P(S)}$  is given by

${\displaystyle P(S)=q^{N_{\text{sample}}(S)}\times (1-q)^{(N_{\text{pop}}-N_{\text{sample}}(S))}}$ 

where for each element ${\displaystyle q}$  is the probability of being included in the sample and ${\displaystyle N_{\text{sample}}(S)}$  is the total number of elements in the sample ${\displaystyle S}$  and ${\displaystyle N_{\text{pop}}}$  is the total number of elements in the population (before sampling commenced).

In business research, companies must often generate samples of customers, clients, employees, and so forth to gather their opinions. Sample design is also a critical component of marketing research and employee research for many organizations. During sample design, firms must answer questions such as:

• What is the relevant population, sampling frame, and sampling unit?
• What is the appropriate margin of error that should be achieved?
• How should sampling error and non-sampling error be assessed and balanced?

These issues require very careful consideration, and good commentaries are provided in several sources.^[1][2]

• Bernoulli sampling
• Sampling probability
• Sampling (statistics)

• Sarndal, Swenson, and Wretman (1992), Model Assisted Survey Sampling, Springer-Verlag, ISBN 0-387-40620-4