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?
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
P(4) = 16c4*(0.267^4)*(1-0.267)^16-4 = 0.22251747806
Fewer than 5
P(0) + P(1) + P(2) + P(3) + P(4) = 0.56847682269
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)
When probability is less than 0.05
Then it is an unusual event
So yes i will be surprised
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