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 10 randomly observed individuals exactly 4 do not cover their mouth when? sneezing?

?(b) What is the probability that among 10 randomly observed individuals fewer than 5 do not cover their mouth when? sneezing?

?(c) Would you be surprised? if, after observing 10 ?individuals, fewer than half covered their mouth when? sneezing? Why?

?(a) The probability that exactly 4 individuals do not cover their mouth is ________.

Answer 1

X ~ Binomial (n,p)

Where, n = 10, p = 0.267

Binomial probability distribution is

P(X) = ⁿC_x p^x (1-p)^n-x

a)

P( X = 4) = ¹⁰C₄ 0.267⁴ 0.733⁶

= 0.1655

b)

P( X < 5) = P( X <= 4)

= P( X = 0) + P( X = 1) + P( X = 2) + P( X = 3) + P( X = 4)  

= ¹⁰C₀ 0.267⁰ 0.733¹⁰ +¹⁰C₁ 0.267¹ 0.733⁹ +¹⁰C₂ 0.267² 0.733⁸ +¹⁰C₃ 0.267³ 0.733⁷+ ¹⁰C₄ 0.267⁴ 0.733⁶

= 0.9004

c)

P(individual covered their mouth when sneezing) = 1 - 0.267 = 0.733

Therefore,

P( Fewer than half covered theire mouth when sneezing) = P( X < 5)

= P( X <= 4)

= P( X = 0) + P( X = 1) + P( X = 2) + P( X = 3) + P( X = 4)  

= ¹⁰C₀ 0.733⁰ 0.267¹⁰ +¹⁰C₁ 0.733¹ 0.267⁹ +¹⁰C₂ 0.733² 0.267⁸ +¹⁰C₃ 0.733³ 0.267⁷+ ¹⁰C₄ 0.733⁴ 0.267⁶

= 0.0272

Since probability is less than 0.05 (that is smaller) we would be surprised if after observing 10

individuals, fewer than half covered their mouth when sneezing.