A study shows that 78 % of the population of all calculus students consider calculus an exciting subject. Suppose 20 students are randomly and independently selected from the population. If the true percentage is really 78 %, find the probability of observing 19 or more of the students who consider calculus to be an exciting subject in the sample of 20. Round to six decimal places.

0.046145 = 4.6145%

Step-by-step explanation:

We need to find the probability of 19 or 20 (all) students considering calculus to be an exciting subject, so first we find the probability for 19 students, then for 20 students, and we sum these probabilities.

The probability of the student NOT considering calculus to be an exciting subject is 1 - 0.78 = 0.22

For 19 students, we also have to calculate a combination of 20 choose 19, as these 19 students can be in different "position" among the 20 students, so:

C (20,19) = 20!/19! = 20

Probability of 19 students considering calculus to be an exciting subject:

C (20,19) * 0.78^19 * 0.22^1 = 0.039196756

Probability of 20 students considering calculus to be an exciting subject:

C (20,20) * 0.78^20 = 1 * 0.78^20 = 0.006948516

The final probability is:

0.039196756 + 0.006948516 = 0.046145272 = 4.6145272%

Rounding to six decimal places: 0.046145 = 4.6145%