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

​(​b)What is the probability that among 16 randomly observed individuals fewer than 5 do not cover their mouth when​ sneezing?

​(​c)Would you be surprised​ if, after observing 16 ​individuals, fewer than half covered their mouth when​ sneezing? Why?

Answer 1

Answer)

As there are fixed number of trials and probability of each and every trial is same and independent of each other

Here we need to use the binomial formula

P(r) = ncr*(p^r)*(1-p)^n-r

Ncr = n!/(r!*(n-r)!)

N! = N*n-1*n-2*n-3*n-4*n-5........till 1

For example 5! = 5*4*3*2*1

Special case is 0! = 1

P = probability of single trial = 0.267

N = number of trials = 16

R = desired success

A)

P(4) = 16c4*(0.267^4)*(1-0.267)^16-4 = 0.22251747806

B)

Fewer than 5

P(0) + P(1) + P(2) + P(3) + P(4) = 0.56847682269

C)

Fewer than half

Here P = 1 - 0.267 = 0.733 (as now they are asking fewer than half covered their mouth)

= P(0) + P(1) + .... P(7)

= 0.0117879153

When probability is less than 0.05

Then it is an unusual event

So yes i will be surprised