Question

According to a study done by a university​ student, the probability a randomly selected individual will not cover his or her mouth when sneezing is 0.2670.

Suppose you sit on a bench in a mall and observe​ people's habits as they sneeze.

​(a) What is the probability that among 18 randomly observed individuals exactly 5 do not cover their mouth when​ sneezing?

​(b) What is the probability that among 18 randomly observed individuals fewer than 3 do not cover their mouth when​ sneezing?

​(c) Would you be surprised​ if, after observing 18 ​individuals, fewer than half covered their mouth when​ sneezing? Why?

Answer 1

We are given here:
P( selected individual will not cover his or her mouth when sneezing ) = 02670

a) The  probability that among 18 randomly observed individuals exactly 5 do not cover their mouth when​ sneezing is computed using binomial probability function as:

Therefore 0.2050 is the required probability here.

b) The probability that among 18 randomly observed individuals fewer than 3 do not cover their mouth when ​sneezing is computed here as:

P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)

Therefore 0.1039 is the required probability here.

c) Probability that fewer than half covered their mouth when​ sneezing
= Probability that more than or equal to 9 do not cover their mouth when​ sneezing

P(X >= 9) is first computed again, using the same method as above parts,
P(X >= 9) = 1 - P(X < 9) = 1 - 0.9706 = 0.0294

As the probability here is 0.0294 < 0.05 , therefore yes, the event here is unusual and we would be surprised to see the event mentioned here.