Question

Asked Jun 21, 2019

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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 15 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*.

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

(c) Find the probability of at least nine coughs (in a large auditorium) in a 32-second period. (Use 4 decimal places.)

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

Coughs are a rare 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 independent.

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

(c) Find the probability of at least nine coughs (in a large auditorium) in a 32-second period. (Use 4 decimal places.)

Step 1

**(a)**

Poisson distribution is most suitable, when the events are “rare” and occur independently of each other.

If an audience is large, 15 coughs in a minute imply a very small number of coughs, as compared to the audience size. Moreover, it can be assumed that the coughs by the different members of the audience are independent of each other.

Thus, in this case, the conditions for a Poisson distribution are satisfied.

The correct option is the fourth option, “**Coughs are a rare occurrence. It is reasonable to assume the events are independent.**”

Step 2

**Computing the probability of 12 or fewer coughs in a minute:**

In the given situation, *r *denotes the number of coughs per minute and *λ* is the average number of coughs per minute.

Thus, *λ* = 15.

The probability of 12 or fewer coughs in a minute is *P*(*r* ≤ 12).

Using EXCEL formula: **=POISSON.DIST(12,15,1)**, the probability of 12 or fewer coughs in a minute is found to be **0.2676**.

Step 3

**(c) Computing the probability of at least 9 coughs in 32 seconds:**

In 1 minute or 60 seconds, average number of coughs is given as *λ* = 15.

Thus, in 32 seconds, the average number of coughs would be *λꞌ* = (15 / 60) * 32 = 8.

The probability of at least 9 coughs in 32 seconds is *P*(*r* ≥ 9) = 1 – *P*(*r* < ...

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