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 0.267. 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 8 do not cover their mouth when​ sneezing? ​(b) what is the probability that among 18 randomly observed individuals fewer than 6 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

 Let X be a discrete binomial random variable.
 Let p = 0.267 be the probability that a person does not cover his mouth when sneezing.
 Let n = 18 be the number of independent tests.
 Let x be the number of successes.
 So, the probability that the 18 individuals, 8 do not cover their mouth after sneezing will be:

 a) P (X = 8) = 18! / (8! * 10!) * ((0.267) ^ 8) * ((1-0.267) ^ (18-8)).
 P (X = 8) = 0.0506.

 b) The probability that between 18 individuals observed at random less than 6 does not cover their mouth is:

 P (X = 5) + P (X = 4) + P (X = 3) + P (X = 2) + P (X = 1) + P (X = 0) = 0.6571.

 c) If it was surprising, according to the previous calculation, the probability that less than 6 people out of 18 do not cover their mouths is 66%. Which means it's less likely that more than half of people will not cover their mouths when they sneeze.