Answer to Question #446 in Statistics and Probability for David

Question

In a certain group of people there is 1% of color blind. How big a random sample should be for the probability of presence of at least one color-blind in it was not less than 0.95?

Answer 1

If there is 1% of color-blind, then among 100 people there will be 1 color-blind person with the probability 1. We have to select P persons out of 100 so that the probability of the presence of a color blind among these P people was not less than 0.95. There are (C100)^P ways to select P persons out of 100. Considered event: there is 1 color-blind and P-1 not color-blind among P selected people. There is (C1)^1=1 way to select a color-blind and there is (C99)^(P-1) ways to select (P-1) not color-blind persons out of (100-1). Thus, the number of options to select P people with 1 color-blind among them is 1*(C99)^(P-1) = (C99)^(P-1). The probability of such an outcome is: (C99)^(P-1)/((C100)^P) = (99!*(100-P)!*P!)/(100!*(100-P)!*(P-1)!) = P/100 = 0.95, thus, P = 95.