We'll use the binomial approximation to the hypergeometric to answer this question. Since we have a fixed number of Americans, a success condition, and a constant probability (from binomial approximation) then we can apply the binomial distribution.
a) Exactly 4: 31C4 (0.17)^4 (1 - 0.17)^27
b) at most 4: P(X <= 4) = P(X = 0) + P(X = 1) + ... + P(X = 4)
For this question, I suggest using a calculator because this is what you must calculate:
= 31C0 (0.17)^0 (1 - 0.17)^31 + 31C1 (0.17)^1 (1 - 0.17)^30 + 31C2 (0.17)^2 (1 - 0.17)^30 + 31C3 (0.17)^3 (1 - 0.17)^29 + 31C4 (0.17)^4 (1 - 0.17)^27
https://stattrek.com/online-calculator/binomial
This is the calculator I use for binomial questions. Plugging 4 into the cumulative distribution function yields a value of 0.3747.
c) At least 6
P(at least 6) = P(X >= 6) = 0.43541 (from calculator)
d) Between 2 and 9
P(Between 2 and 9) = P(2 <= X <= 9). Using the calculator above, what we can do is find P(X <= 9) and subtract off the probability of being less than or equal to 2.
P(2 <= X <= 9) = P(X <= 9) - P(X <= 2) = 0.97181 - 0.08327 = 0.88854
Hope this helps!
