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# 2.2.1. Probability of pathogen detection in a colony based on a known sample size

Instead of focusing on sample size, one can calculate the resulting probability of detection of a disease organism using a specific sample size. This probability can be calculated (for an individual colony) using Equation I, but solving for D, as given in Equation II, below.

Equation II. D = 1-(1-P)N

where D, P, and N are defined as in Equation I above.

For example, within a colony, if the pathogen prevalence in worker bees is 10% (90% of bees are non-infected), then the probability of detecting the pathogen in the colony using a sample size of one bee is 0.10, much lower than that for 30 bees (probability is 0.96). A lower prevalence will lower the probability of detection for the same sample size (Fig. 2).

Based on Equation II, it is also possible to calculate the number of bees that need to be tested (sample size) to detect at least one infected bee as a function of the probability, e.g. at a probability of detection (D), of 95% or 99% (Fig. 3). The number bees to be tested to detect at least one infected bee is higher if one needs a higher probability of detection (D), i.e. when one needs to be able to detect low prevalence.

Fig. 2. The probability of detecting a pathogen in a colony (D) as a function of the sample size of bees from that colony, where bees are a completely random sample from the colony.  The minimal (true) infection prevalences (P) are 10% (solid line), 5% (dashed line), and 1% (dotted line).

Fig. 3. The number of bees that need to be tested (sample size) to detect at least one infected bee as a function of the prevalence (P), e.g. at a probability of detection (D) of 95% (solid line) or 99% (striped line).