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 12 randomly observed individuals exactly 8 do not cover their mouth when​ sneezing? ​(b) What is the probability that among 12 randomly observed individuals fewer than 6 do not cover their mouth when​ sneezing? ​(c) Would you be surprised​ if, after observing 12 ​individuals, fewer than half covered their mouth when​ sneezing? Why?

Answer 1

Answer: kindly check explanation

Step-by-step explanation:

Probability of not covering mouth = p(success) = 0.267

Hence, p(covering mouth) = 1 - 0.267 = 0.733

a) What is the probability that among 12 randomly observed individuals exactly 8 do not cover their mouth when​ sneezing? ​

Number of samples (n) = 12

P(X = 8)

Using binomial distribution :

P(x) = nCx * p^x * (1-p)^(n-x)

P(x =8) = 12C8 * 0.267^8 * 0.733^(12-8)

P(x =8) = 12C8 * 0.267^8 * 0.733^4

= 0.00369

b) What is the probability that among 12 randomly observed individuals fewer than 6 do not cover their mouth when​ sneezing?

P(X < 6) = p(5) + p(4) + p(3) + p(2) + p(1) + p(0)

To save computation time, we can use an online binomial probability calculator :

P(X < 6) = 0.9275

C.) Yes, I will be surprised, because from the binomial probability obtained above, there is a high probability (0.9275) that fewer than half do not cover their mouth.