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.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.)

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Asked Jun 22, 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.
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.)
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Step 1

Given that in any large audience you will hear that 15 coughs per minute. r is the number of coughs in a given time interval.
Coughs are rare occurrence and it is reasonabile to assume that events are independent. Here the variable is number of coughs in a given time interval, so it is good choice to use poisson distribution for probability distribution of r.

Step 2

The variable r has poisson distribution with λ = 15. The probability distribution function is shown below.

The probability that 12 or fewer coughs in a 1-minut...

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e N 7 P(r,A) r! 8.5 7.0 10.0 10.0 12.0 14.0 15.0 6.0 6.5 7.5 8.0 9.0 9.5 11.0 0.0025 0.0015 0.0009 0.0006 0.0003 0.0002 0.0001 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0174 0.0113 0.0073 0.0047 0.0030 0.0019 0.0012 0.0008 0.0005 2 0.0620 0.0430 0.0296 0.0203 0.0138 0.0093 0.0062 0.0042 0.0028 0.0012 0.0028 0.0005 0.0001 0.0000 3 0.1512 0.1118 0.0818 0.0591 0.0424 0.0301 0.0212 0.0149 0.01 03 0.0049 0.0103 0.0023 0.0005 0.0002 0.0002 0.0005 0.0001 0.0000 0.0000 1 4 0.2851 0.2237 0.1730 0.1321 0.0996 0.0744 0.0550 0.0403 0.0293 0.0151 0.0293 0.0076 0.0018 0.0009 5 0.4457 0.3690 0.3007 0.2414 0.1912 0.1496 0.1157 0.0885 0.0671 0.0375 0.0671 0.0203 0.0055 0.0028 0.6063 0.5265 0.4497 0.3782 0.31 34 0.2562 0.2068 0.1649 0.1301 0.7440 6 0.0786 0.1301 0.0458 0.0142 0.0076 0.0895 0.0316 0.0180 0.6728 0.4530 7 0.5987 0.5246 0.3856 0.3239 0.2687 0.2202 0.1432 0.2202 8 0.8472 0.7916 0.7291 9 0.9161 10 0.9574 0.9332 0.9015 0.8622 0.8159 0.7634 0.7060 0.6620 0.5925 0.5231 0.4557 0.3918 0.3328 0.2320 0.3328 0.1550 0.0621 0.0374 0.8774 0.8305 0.7764 0.7166 0.6530 0.5874 0.5218 0.4579 0.3405 0.4579 0.2424 0.1094 0.0699 0.6453 0.5830 0.4599 0.5830 0.3472 0.1757 0.1185 0.9467 0.9208 0.8881 0.8487 0.8030 0.7520 0.6968 0.5793 0.6968 0.4616 0.2600 0.1848 12 0.9912 0.9840 0.9730 0.9573 0.9362 0.9091 0.8758 0.8364 0.7916 0.6887 0.7916 0.5760 0.3585 0.2676 13 0.9964 0.9929 0.9872 0.9784 0.9658 0.9486 0.9261 0.8981 0.8645 0.781 3 0.8645 0.6815 0.4644 0.3632 14 0.9986 0.9970 0.9943 0.9897 0.9827 0.9726 0.9585 0.9400 0.9165 0.8540 0.9165 0.7720 0.5704 0.4657 11 0.9799 0.9661

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