If it rains in Seattle 75% of all days and I check the weather there for a sample of n=30 days what is the probability that it will rain for more than 20 of those days?
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p=0.75 is the probability that it will rain on any particular day; 1-0.75=0.25 is the probability that it won't rain. There are nCr, where n=30 and r=20, ways of picking 20 out of 30 days=30,045,015 ways, combining the probabilities of 20 days of rain with 10 days of no rain, i.e., (p^20)*(1-p)^10=0.75^20*0.25^10=3.0243*10^-9. Multiply these two factors together and we get 0.0909 or 9.09% probability that it will rain 20 days out of 30. (Binomial distribution assumed.)

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