Exam P Practice Problem 8

Problem 8a
Suppose that the number of parking tickets issued in a calendar year to a certain driver follows a Poisson distribution with mean of 4.7. The fine for each ticket is $75. What is the probability that this driver will pay more than $350 in fine in the upcoming calendar year?

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Problem 8b
Suppose a certain tour operator offers excursion tours and the number of tourists served by this tour operator follows a Poisson distribution with mean of 12 per day. The cost of the excursion tour is $125 per person. What is the probability that the total daily revenue will not exceed $700?

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Solution is found below.

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Solution to Problem 8a
The probability function of the Poisson distribution in question is:

\displaystyle P(N=n)=\frac{e^{-4.7} 4.7^n}{n!}

Having 5 or more tickets would lead to a total fine of more than $350. Thus the answer is:

\displaystyle \begin{aligned}P(N \ge 5)&=1-P(N \le 4) \\&=1-e^{-4.7}\biggl(1+4.7+\frac{4.7^2}{2!}+\frac{4.7^3}{3!}+\frac{4.7^4}{4!}\biggr) \\&=1-54.3808375 e^{-4.7} \\&=0.5053912139   \end{aligned}

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Answer to Problem 8b
0.0203410294

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