Fifteen Dolphins in the Baltic Sea
Suppose that, owing to prevailing conditions during AD 3001 in the Baltic Sea, the probability that a female dolphin Carla who is swimming in the Gulf of Finland has the ‘gay’ phenotype Ω is 0.5, and that the conditional probability that any particular daughter has phenotype Ω given that Carla, the mother, has the phenotype Ω, is equal to 0.5 . Then according the probability that both Carla her daughter Zena have phenotype Ω is 0.5x0.5=0.25.
Suppose however that Carla has 14 daughters and consider the event A that Carla and all 14 of her daughters have phenotype Ω. It would be a mistake to obtain prob (A) by multiplying 0.5 by 0.25 to the power 14, giving prob (A)=1.8626 times one in a billion. Since the 15 constituent events are clearly interconnected and not independent, we are not permitted to multiply the unconditional probabilities together.
In order to proceed, we make the further (in itself highly tenuous) assumption, that conditionally on Carla having phenotype Ω, the 14 events that her 14 daughters have gay phenotype Ω are mutually independent. Under this conditional independence assumption, prob (A) = 0.5 x (0.5 to the power 14) =0.00003052. This multiplies the previous miscalculated number by a factor of 2 to the power 14, which equals 16384. Such extremely severe problems abound when trying to calculate the probabilities of any intersection of events in genetics, particularly as the events may be interconnected in all sorts of indecipherable ways.
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