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 18 coughs per minute.

(b) Find the probability of four 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 32-second period. (Use 4 decimal places.)

Answer 1

Solution:

To obtain the probability we shall use poisson distribution.

According to poisson process, the probability of occurrence of exactly x events in time t is given by,

Where,  λ is rate of occurrence of the event.

Let X be a random variable which represents the number of coughs in a minute in any large audience.

We shall consider that X has poisson distribution with mean  λt.

We have,  λ = 18 coughs per minute

a) We have to obtain P(X ≤ 4) in t = 1 minute.

We have, λt = 18 × 1 = 18

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

Using poisson distribution we get,

[Since, (x^{​​​​​​0}) = 1, (x^{​​​​​​1}) = x, 0! = 1, 1! = 1 and n! = n×(n-1)×(n-2)×..........×1 ]

On rounding to 4 decimal places we get,

P(X ≤ 4) = 0.0001

The probability of four or fewer coughs (in a large auditorium) in a 1-minute period is 0.0001.

b) We have to obtain P(X ≥ 8) in t = 32 seconds.

32 seconds = 32/60 minutes

Hence, λt = 18 × (32/60) = 9.6

P(X ≥ 8) = 1 - P(X < 8)

P(X ≥ 8) = 1 - [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7)]

On rounding to 4 decimal places we get,

P(X ≥ 8) = 0.7416

The probability of at least eight coughs (in a large auditorium) in a 32-second period is 0.7416.

Please rate the answer. Thank you.