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Expand Your Knowledge: Conditional Probability Pyramid Lake is located in Nevada on the Paiute Indian Reservation. This lake is famous for large cutthroat trout. The mean number of trout (large and small) caught from a boat is \(0.667\) fish per hour (Reference: Creel Chronicle, Vol. 3, No. 2 ). Suppose you rent a boat and go fishing for 8 hours. Let \(r\) be a random variable that represents the number of fish you catch in the 8 -hour period. (a) Explain why a Poisson probability distribution is appropriate for \(r\). What is \(\lambda\) for the 8 -hour fishing trip? Round \(\lambda\) to the nearest tenth so that you can use Table 4 of Appendix II for Poisson probabilities. (b) If you have already caught three trout, what is the probability you will catch a total of seven or more trout? Compute \(P(r \geq 7 \mid r \geq 3) .\) See Hint below. (c) If you have already caught four trout, what is the probability you will catch a total of fewer than nine trout? Compute \(P(r<9 \mid r \geq 4) .\) See Hint below. (d) List at least three other areas besides fishing to which you think conditional Poisson probabilities can be applied. Hint for solution: Review item 6 , conditional probability, in the summary of basic probability rules at the end of Section \(4.2 .\) Note that $$ P(A \mid B)=\frac{P(A \text { and } B)}{P(B)} $$ and show that in part (b), $$ P(r \geq 7 \mid r \geq 3)=\frac{P((r \geq 7) \text { and }(r \geq 3))}{P(r \geq 3)}=\frac{P(r \geq 7)}{P(r \geq 3)} $$

Step by step solution

01

Justification for Poisson Distribution

A Poisson distribution is appropriate because we are counting the number of events (trout catches) occurring over a continuous interval (time) with a known average rate (0.667 trout per hour). Events are also assumed to be independent. Therefore, a Poisson distribution can model this situation.

02

Calculate Poisson Parameter \(\lambda\)

We know the mean number of fish caught is 0.667 per hour. For an 8-hour period, the expected number (\(\lambda\)) is computed as \(\lambda = 8 \times 0.667 = 5.336\). Rounding 5.336 to the nearest tenth gives us \(\lambda = 5.3\).

03

Conditional Probability for Seven or More Fish After Three

We want to calculate \(P(r \geq 7 \mid r \geq 3)\). According to the conditional probability formula, it's \(\frac{P(r \geq 7)}{P(r \geq 3)}\). First, we find \(P(r \geq 7)\) and \(P(r \geq 3)\) using Poisson probability tables or calculations with \(\lambda = 5.3\).

04

Use Poisson Probability for \(P(r \geq 7)\) and \(P(r \geq 3)\)

Using a Poisson table or calculator, \(P(r \geq 7)\) is calculated by finding 1 minus the sum of probabilities for catching 0 to 6 fish. \(P(r \geq 3)\) is found by subtracting the sum of probabilities from 0 to 2 fish from 1. Plug these into the conditional formula for the result.

05

Conditional Probability for Fewer Than Nine Fish After Four

To find \(P(r < 9 \mid r \geq 4)\), we use \(\frac{P(r < 9 \text{ and } r \geq 4)}{P(r \geq 4)}\). Here, \(P(r < 9 \text{ and } r \geq 4) = P(r < 9) - P(r < 4)\), and \(P(r \geq 4) = 1 - P(r < 4)\).

06

List Other Applications of Conditional Poisson Probabilities

Poisson probabilities can be applied to other situations like counting the number of phone call arrivals at a call center during a time period, calculating the number of typing errors made by a typist in a manuscript, or counting decay events of radioactive substance in an hour.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Poisson Distribution

In statistics, the Poisson distribution is used to model the number of times an event occurs in a fixed interval of time or space. The key features of a Poisson distribution involve:

• The event is something that can be counted in whole numbers.
• Occurrences are independent; one event does not affect the probability of another.
• The average rate of occurrence is constant throughout the observation period.
• Two events cannot occur at exactly the same instance.

In our fishing example, the Poisson distribution suits perfectly as we are counting the number of trout caught over a period of time (in hours), with a constant average catch rate of 0.667 trout per hour. The randomness and independence of catches further validate the use of Poisson distribution. Additionally, we calculate \[\lambda = 8 \times 0.667 = 5.336,\]and rounding gives us \[\lambda = 5.3,\]which is used to compute various probabilities.

Random Variables

A random variable is a numerical description of the outcome of a statistical experiment. In simpler terms, it's a variable that can take on different values, each with an associated probability, depending on the outcome of a particular event. There are two main types of random variables:

• **Discrete Random Variable:** Takes on countable values, like the number of fish caught.
• **Continuous Random Variable:** Takes on infinite values within a given range, like the exact time you might catch a fish.

In the fishing context, the random variable \(r\) represents the number of fish caught during an 8-hour session, which is a discrete random variable because you can count distinct outcomes (0, 1, 2, 3, etc.). It's these outcomes that we use in calculations involving Poisson probabilities.

Probability Distributions

Probability distributions describe how the probabilities of different outcomes are spread out or distributed. They provide a framework for understanding the likelihood of different results in an experiment.The Poisson probability distribution is particularly useful when it comes to modeling events with a known average rate and are randomly occurring over time. For our exercise, once we have determined the correct \(\lambda\),we use the probability distribution to determine specific probabilities like \(P(r \geq 7)\)and \(P(r \geq 3)\).These probabilities help calculate conditional probabilities such as \(P(r \geq 7 \mid r \geq 3)\),which tells us about the likelihood of catching 7 or more fish given that 3 have already been caught.Probability distributions simplify complex random processes into manageable statistical problems, thus allowing us to predict outcomes more effectively.