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Just a note
The binomial expansion is a formula which calculates the probability of obtaining p successes in a sample of n observations from a population, assuming observations are independent and random, and the probability of observing a success is P  and that random sampling is the only source of variation.
After obtaining these results we multiplied the probability of observing each result, given by the binomial expansion, by the number of observations  and subjected p to an arcsin square root transformation.
