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Ch. 5 - Discrete Probability Distributions
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 5 - Discrete Probability DistributionsProblem 5.R.10b
Chapter 5, Problem 5.R.10b

Poisson: Deaths Currently, an average of 7 residents of the village of Westport (population 760) die each year (based on data from the U.S. National Center for Health Statistics).


b. Find the probability that on a given day, there are no deaths.

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Step 1: Recognize that this is a Poisson probability problem. The Poisson distribution is used to model the number of events (e.g., deaths) occurring in a fixed interval of time or space, given a known average rate (λ). Here, the average rate is 7 deaths per year.
Step 2: Convert the average rate (λ) from yearly to daily. Since there are 365 days in a year, divide the yearly rate by 365 to find the daily rate: λ = 7 / 365.
Step 3: Use the Poisson probability formula to calculate the probability of 0 deaths on a given day. The formula is: P(X = k) = (λ^k * e^(-λ)) / k!, where k is the number of events (in this case, k = 0), λ is the average rate, and e is the base of the natural logarithm (approximately 2.718).
Step 4: Substitute k = 0 and the daily rate λ (calculated in Step 2) into the formula. Simplify the expression: P(X = 0) = (λ^0 * e^(-λ)) / 0!. Note that 0! = 1 and λ^0 = 1, so the formula simplifies to P(X = 0) = e^(-λ).
Step 5: Compute e^(-λ) using the daily rate λ. This will give you the probability of no deaths occurring on a given day.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Poisson Distribution

The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given a known average rate of occurrence. It is particularly useful for modeling rare events, such as deaths in a small population, where the events occur independently of each other.
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Intro to Frequency Distributions

Rate Parameter (λ)

In the context of the Poisson distribution, the rate parameter (λ) represents the average number of occurrences in the specified interval. For the village of Westport, with an average of 7 deaths per year, λ would be 7. When calculating probabilities for shorter intervals, such as a day, λ must be adjusted accordingly (e.g., λ = 7/365 for daily calculations).
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Probability of No Events

To find the probability of observing no events in a Poisson distribution, the formula P(X=0) = e^(-λ) is used, where e is the base of the natural logarithm. This formula indicates the likelihood of zero occurrences when the average rate is λ. In this case, it helps determine the probability of no deaths occurring on a given day in Westport.
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Related Practice
Textbook Question

Poisson: Deaths Currently, an average of 7 residents of the village of Westport (population 760) die each year (based on data from the U.S. National Center for Health Statistics).


c. Find the probability that on a given day, there is more than one death.


Textbook Question

In Exercises 1–5, assume that 4.2% of workers test positive when tested for illegal drugs (based on data from Quest Diagnostics). Assume that a group of ten workers is randomly selected.


Workplace Drug Testing Find the mean and standard deviation for the numbers of workers in groups of ten who test positive for illegal drugs.

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Textbook Question

In Exercises 1–5, assume that 4.2% of workers test positive when tested for illegal drugs (based on data from Quest Diagnostics). Assume that a group of ten workers is randomly selected.


Workplace Drug Testing If four of the ten workers test positive for illegal drugs, is that a significantly high result?

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Textbook Question

Family/Partner Groups of people aged 15–65 are randomly selected and arranged in groups of six. The random variable x is the number in the group who say that their family and/or partner contribute most to their happiness (based on a Coca-Cola survey). The accompanying table lists the values of x along with their corresponding probabilities. Does the table describe a probability distribution? If so, find the mean and standard deviation.


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Textbook Question

In Exercises 1–5, assume that 4.2% of workers test positive when tested for illegal drugs (based on data from Quest Diagnostics). Assume that a group of ten workers is randomly selected.


Workplace Drug Testing Find the probability that at least one of the ten workers tests positive for illegal drugs.

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Textbook Question

Acrophobia USA Today reported results from a survey in which subjects were asked if they are afraid of heights in tall buildings. The results are summarized in the accompanying table. Does this table describe a probability distribution? Why or why not?

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