# 2.2.2. Probability of pathogen detection in a population of colonies

If one wants to calculate the probability of detection of a pathogen in a population of colonies using a known number of colonies, this probability can be calculated (for a population of colonies) according to Equation III.

Equation
III. E = 1-(1-P*D)^{N}

where

E = the probability of detection (in the population)

D and P are defined as above in Equation I and II, N is the sample size, in this case number of colonies

If one
wants to determine if a pathogen is present in a population and the probability
of pathogen detection in individual colonies is known (see Equation II above),
then one can calculate how many colonies need to be sampled in order to detect
that pathogen using Equation IV. The computation now calculates the probability
of *at least* one positive recording in
two-stage sampling situations (the probability of detection in individual
colonies and the probability of detection in the population). The probability
of pathogen detection in the population can be calculated using Equation III,
or it can be set to 0.95 or 0.99, depending on the power required for the
investigation.

Equation IV. N = ln(1-E) / ln(P * D)

where

N, ln, E, P, and D are defined as above in Equations I, II and III

Equation I, II, III, and IV can easily be entered into a spread sheet for calculation of sample sizes needed for different purposes and desired probabilities of detection.