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 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

X ~ Binomial (n,p)

Where n = 18 , p = 0.267 (Probability of do not cover)

For binomial distribution,

P(X) = ⁿC_x p^x ( 1 - p)^n-x

a )

P( X = 8) = ¹⁸C₈ 0.267⁸ 0.733¹⁰

= 0.0506

b)

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

= ¹⁸C₀ 0.267⁰ 0.733¹⁸ +¹⁸C₁ 0.267¹ 0.733¹⁷ +¹⁸C₂ 0.267² 0.733¹⁶ +¹⁸C₃ 0.267³ 0.733¹⁵

= 0.2511

c)

p(Covered their mouth when sneezing) = 1 - 0.267 = 0.733

P( X < 9) = P( X <= 8)

= P(X = 0) +P(X = 1) +P(X = 2) +P(X = 3) +P(X = 4) +P(X = 5) +P(X = 6) +P(X = 7) +P(X = 8)

= ¹⁸C₀ 0.733⁰ 0.267¹⁸ +¹⁸C₁ 0.733¹ 0.267¹⁷ +¹⁸C₂ 0.733² 0.267¹⁶ +¹⁸C₃ 0.733³ 0.267¹⁵

+ ¹⁸C₄ 0.733⁴ 0.267¹⁴ +¹⁸C₅ 0.733⁵ 0.267¹³ +¹⁸C₆ 0.733⁶ 0.267¹² +¹⁸C₇ 0.733⁷ 0.267¹¹

+ ¹⁸C₈ 0.733⁸ 0.267¹⁰

= 0.0089

This probability is less than 0.05. So the event fewer than half covered their mouth while sneezing

is unusual.

Therefore,

Yes. We would surprised if fewer than half covered thier mouth while sneezing.