Answer:
A
[tex]P(X = 8 ) = 0.0037[/tex]
B
[tex]P(X < 5) = 0.805[/tex]
C
I will not be surprised because the probability that fewer than half covered their mouth when sneezing is less than 0.5
Step-by-step explanation:
From the question we are told that
The probability a randomly selected individual will not cover his or her mouth when sneezing is [tex]p = 0.267[/tex]
The probability a randomly selected individual will cover his or her mouth when sneezing is
[tex]q = 1 -p[/tex]
[tex]q = 1 -0.267[/tex]
[tex]q = 0.733[/tex]
Generally the probability that among 12 randomly observed individuals exactly 8 do not cover their mouth when sneezing is mathematically represented as
[tex]P(X = 8 ) = \left 12 } \atop {}} \right. C_8 * p^8 * q^{12-8}[/tex]
[tex]P(X = 8 ) = 495 * (0.267)^8 * (0.733)^{12-8}[/tex]
[tex]P(X = 8 ) = 0.0037[/tex]
Generally the probability that among 12 randomly observed individuals fewer than 5 do not cover their mouth when sneezing is mathematically represented as
[tex]P(X < 5 ) = P[ P(X = 0) + \cdots + P(X = 4)][/tex]
=> [tex]P(X < 5) = \left 12 } \atop {}} \right. C_0 * (0.267)^0 * (0.733)^{(12- 0) }+\cdots + \left 12 } \atop {}} \right. C_4 * (0.267)^0 * (0.733)^{(12- 4) }[/tex]
=> [tex]P(X < 5) = 0.805[/tex]
Give that half of 12 is 6 then
The probability that fewer than half covered their mouth when sneezing is mathematically represented as
[tex]P(X > 6) = 1 - P( X \le 6 )[/tex]
=> [tex]P(X > 6) = 1 - [P(X = 0 ) +\cdots+P(X = 6) ][/tex]
=> [tex]P(X > 6) = 1 - [\left 12 } \atop {}} \right. C_0 *(0.267)^0 * (0.733)^{12 - 0 } +\cdots + \left 12 } \atop {}} \right. C_6 * (0.267)^6 * (0.733)^{12-6}][/tex]
=> [tex]P(X > 6) = 0.0206[/tex]
