Exam 12: Probabilistic

A tax firm company selects a small portion of their tax returns to audit. These selected returns are given to auditors in lot sizes of 30. From this group of 30, the auditor randomly selects 10 and rigorously reviews them. Returns that fail this inspection are sent to a special office which then follows up on the investigation. In the past, 5 out of every 30 returns result in a forma audit. What is the probability that in the next 10 returns sampled, only one will need further auditing?

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Question 1

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$\begin{array} { l } N = 30 , D = 5 , n = 10 , d = 1 \\\end{array}$

At a major airport, over 1000 planes land each day on their runways. This steady stream of air traffic has crated complexities and delays for landing and departing planes. At any given moment, the airport managers estimate that there is a 1% chance of a runway incursion (near-miss). What is the probability that in the next sample of 25 flights, 1 or fewer runway incursions will occur?

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Question 2

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$\begin{array}{l}p=0.01, n=25 \\P(1 \text { or fewer })=P(0)+P(1)\end{array}$
$=\frac{25 !}{0 ! 25 !}\left(0.01 \rho(0.99)^{25}+\frac{25 !}{1 ! 24 !}(0.01)^{1}(0.99)^{24}\right.$
$\begin{array}{l}=0.778+0.196 \\=0.974\end{array}$

An environmental engineer places monitors in a large number of streams to measure the amount of pollutants in the water. If the amount of pollutants exceeds a certain level, the water is considered non-conforming. In the past, the proportion non-conforming has been 0.04. What is the probability that one of the 15 samples taken today will contain an excessive amount of pollutants?

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Question 3

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$\begin{array} { l } p = 0.04 , n = 15 \\P ( 1 ) = - \frac { 15 ! } { 1 ! 14 ! } ( 0.04 ) ^ { 1 } ( 0.96 ) ^ { 14 } \\P ( 1 ) = 0.34\end{array}$

On a pipeline, a device know affectionately as a pig, checks the pipeline seams for leaks. The hundreds of thousands of seams along the 500 mile pipeline can be considered a steady stream of product. At any given moment, the pipeline managers estimate that 1% of the seams will need to be repaired. What is the probability that in the next sample of 25 seams, 2 or fewer of the seams will need repair?

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Question 4

Use the Poisson distribution to compute the probability of finding one or fewer paint flaws on the surface finish in a sample of 25 cars, when np = 0.25.

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Question 5

A radio station is hosting a promotional activity at a local mall. In the past, on average, six (np = 6) excited participants wanting to take part in the promotion arrive during any given minute. Since keeping people waiting affects the success of the promotional activity, the radio station wants to staff their booth with enough people to handle this volume. One of the questions they have asked themselves is: what is the probability that the number of participants arriving per minute will be greater than 8?

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Question 6

The number of oil tanker ships arriving at a particular port each day is 10. The facilities at the port can dock and load/unload at most, 15 tanker ships per day. What is the probability that the port will need to turn a ship away? In other words, what is the probability that the number of ships arriving per day will be greater than 15?

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Question 7

A rather harried father is trying to find a much sought after toy for his daughter for Christmas. He has a choice of 10 stores to go to. From a radio announcement, he learned that the toy can definitely be found at 4 of the ten stores. Unfortunately, he did not hear which four stores! If the store he stops at does not have the toy, he will leave without buying anything, in other words, there is no replacement. He only has time to go to three stores. What is the probability that he will find the toy at the first stop?

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Question 8

A company ships 4,600 car axles a day ($1,175 worth of components in each vehicle). This steady stream of production has a defect rate of 0.01%. Calculate the probability that 0 defects will be found in a sample of 160.

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Question 9

Use the Poisson distribution to compute the probability of finding one paint flaw on the surface finish in a sample of 25 cars, when np = 0.25.

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Question 10

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Use the Poisson distribution to compute the probability of finding one or fewer paint flaws on the surface finish in a sample of 25 cars, when np = 0.25.

A company ships 4,600 car axles a day ($1,175 worth of components in each vehicle). This steady stream of production has a defect rate of 0.01%. Calculate the probability that 0 defects will be found in a sample of 160.

Use the Poisson distribution to compute the probability of finding one paint flaw on the surface finish in a sample of 25 cars, when np = 0.25.

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