We believe that 90% of the population of all calculus i students consider calculus an exciting subject. suppose we randomly and independently selected 21 students from the population. if the true percentage is really 90%, find the probability of observing 20 or more of the students who consider calculus to be an exciting subject in our sample of 21.

This is a binomial distribution problem. The formula to find the required probability is:

p(X) = [ n! / ((n - X)! · X!) ] · (p)ˣ · (q)ⁿ⁻ˣ 

where:
X = number of what you are trying to find the probability for = 20 or 21;
n = number of events randomly selected = 21;
p = probabiity of sucess = 0.9 (90%);
q = probability of failure = 0.1 (10%);

We need to find the probability of two events: finding 20 students and finding 21 students. Therefore P(X) = P(X = 20) + P(X = 21).

P(X = 20) =  [ 21! / ((21 - 20)! · 20!) ] · (0.9)²⁰ · (0.1)²¹⁻²⁰ 
                =  0.2553

P(X = 21) = [ 21! / ((21 - 21)! · 21!) ] · (0.9)²¹ · (0.1)²¹⁻²¹ 
                = 0.1094

Therefore:
P(X) = 0.2553 + 0.1094 = 0.3647

We have a probability of 36.5% to find 20 or more students who consider calculus to be an exciting subject.