Question

A person with a cough is a persona non grata on airplanes, elevators, or at the theater. In theaters especially, the irritation level rises with each muffled explosion. According to Dr. Brian Carlin, a Pittsburgh pulmonologist, in any large audience you'll hear about 8 coughs per minute.

(a) Let r = number of coughs in a given time interval. Explain why the Poisson distribution would be a good choice for the probability distribution of r.

Coughs are a common occurrence. It is reasonable to assume the events are dependent.Coughs are a common occurrence. It is reasonable to assume the events are independent.    Coughs are a rare occurrence. It is reasonable to assume the events are dependent.Coughs are a rare occurrence. It is reasonable to assume the events are independent.

(b) Find the probability of six or fewer coughs (in a large auditorium) in a 1-minute period. (Use 4 decimal places.)


(c) Find the probability of at least eight coughs (in a large auditorium) in a 24-second period. (Use 4 decimal places.)

Answer 1

a) Coughs are a rare occurrence. It is reasonable to assume the events are independent.

Let r = number of coughs in a given time interval.

r follows Poisson distribution with λ = 8

P(r) =   , r = 0,1,2,3...

b) P( r ≤ 6 ) = P(0) + P(1) +P(2) +P(3) +P(4) +P(5)+ P(6)

=   + + + + +  +

= 0.0003 + 0.0027 + 0.0107 +0.0286 +0.0573 +0.0916 + 0.1221

= 0.3133

The probability of six or fewer coughs (in a large auditorium) in a 1-minute period is 0.3133

c) P( r ≥ 8 ) = 1 - P( r ≤ 7 )

= 1 - [ P(0) + P(1) +P(2) +P(3) +P(4) +P(5) P(6) +P(7) ]

= 1 - [    + + + + +  ++ ]

= 1 - [ 0.0003 + 0.0027 + 0.0107 +0.0286 +0.0573 +0.0916 + 0.1221 + 0.1396 ]

= 1 - 0.4529

= 0.5471

The probability of at least eight coughs (in a large auditorium) in a 24-second period is 0.5471