This calculator uses Bayes’ theorem to calculate the effects of different series of observations on the probability of a hypothesis. Observation reports can take the two forms

so it is natural to think of the hypothesis in question as All Ss are P. However, the specific form of the hypothesis plays a role in the calculations only by way the initial (i.e., “prior”) probabilities assigned to it and assigned to observation reports relative to it and its denial. The textboxes showing initial probabilities begin with sample values (which you can change). To makes things all little more concrete, All ravens are black is chosen as the hypothesis H.

Repeated observations of black ravens should raise the probability that the next raven to be observed will be black even when it is assumed that not all ravens are black. To handle this, the alternative hypothesis not all ravens are black is partitioned into a number of more specific hypotheses concerning the proportion of ravens that are black. These will have their probability altered by repeated observations and that will alter the probability of the more general alternative hypothesis. The finer the partition the better, but extremely large numbers can be slow to calculate, so you can set the number considered. (The default setting is actually rather low relative to what recent computers and browsers can handle, so you might try raising it.)

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Inputs

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Prior probabilities

prob(H) =	prob(not-H) =
prob(BR given H) =	prob(BR given not-H) =
prob(nBR given H) =	prob(nBR given not-H) =

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Posterior probabilities

Number of observations:   black ravens:   non-black ravens:   prob(H given …):   prob(BR given H and …): 1 prob(BR given not-H and …):   prob(BR given …):