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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 5.3.9b
Chapter 5, Problem 5.3.9b

In Exercises 9–16, use the Poisson distribution to find the indicated probabilities.


Births In a recent year (365 days), NYU-Langone Medical Center had 5942 births.


b. Find the probability that in a single day, there are 16 births.

Verified step by step guidance
1
Step 1: Understand the Poisson distribution. The Poisson distribution is used to model the probability of a given number of events (e.g., births) occurring in a fixed interval of time or space, given a known average rate of occurrence (λ). The probability mass function is given by: P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the average rate, k is the number of events, and e is the base of the natural logarithm (approximately 2.718).
Step 2: Calculate the average number of births per day (λ). Since there are 5942 births in 365 days, divide the total number of births by the number of days to find the daily average: λ = 5942 / 365.
Step 3: Identify the value of k. In this problem, k represents the number of births in a single day, which is given as 16.
Step 4: Substitute the values of λ and k into the Poisson probability formula. Use the formula P(X = k) = (λ^k * e^(-λ)) / k!. Replace λ with the calculated daily average and k with 16.
Step 5: Simplify the expression to compute the probability. First, calculate λ^k, e^(-λ), and k! (16 factorial). Then, divide the product of λ^k and e^(-λ) by k! to find the probability. This will give you the final result.

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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 that these events happen with a known constant mean rate and independently of the time since the last event. It is particularly useful for modeling rare events, such as the number of births in a day.
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Mean (λ) in Poisson Distribution

In the context of the Poisson distribution, the mean (denoted as λ, lambda) represents the average number of occurrences of the event in the specified interval. For the given problem, λ would be calculated by dividing the total number of births (5942) by the number of days (365), which gives the expected number of births per day.
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Calculating Probability with Poisson

To find the probability of observing exactly k events (in this case, 16 births) in a Poisson distribution, the formula P(X = k) = (e^(-λ) * λ^k) / k! is used, where e is the base of the natural logarithm, λ is the mean number of events, and k! is the factorial of k. This formula allows us to compute the likelihood of a specific number of occurrences given the average rate.
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Related Practice
Textbook Question

In Exercises 29 and 30, assume that different groups of couples use the XSORT method of gender selection and each couple gives birth to one baby. The XSORT method is designed to increase the likelihood that a baby will be a girl, but assume that the method has no effect, so the probability of a girl is 0.5.


Gender Selection Assume that the groups consist of 36 couples.


b. Use the range rule of thumb to find the values separating results that are significantly low or significantly high.

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Textbook Question

Binomial Probability Formula. In Exercises 13 and 14, answer the questions designed to help understand the rationale for the binomial probability formula.


Guessing Answers Standard tests, such as the SAT, ACT, or Medical College Admission Test (MCAT), typically use multiple choice questions, each with five possible answers (a, b, c, d, e), one of which is correct. Assume that you guess the answers to the first three questions.


b. Beginning with WWC, make a complete list of the different possible arrangements of two wrong answers and one correct answer, and then find the probability for each entry in the list.

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Textbook Question

Politics The County Clerk in Essex, New Jersey, was accused of cheating by not using randomness in assigning the order in which candidates’ names appeared on voting ballots. Among 41 different ballots, Democrats were assigned the desirable first line 40 times. Assume that Democrats and Republicans are assigned the first line using a method of random selection so that they are equally likely to get that first line.


b. Find the probability of exactly 40 first lines for Democrats.

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Textbook Question

Using Probabilities for Significant Events


b. Find the probability of getting 1 or fewer matches.

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Textbook Question

 In Exercises 5–8, assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is 5.5 per year, as in Example 1; and proceed to find the indicated probability.


Hurricanes


b. In a 118-year period, how many years are expected to have 7 hurricanes?

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Textbook Question

In Exercises 5–8, assume that the Poisson distribution applies; assume that the mean number of Atlantic hurricanes in the United States is 5.5 per year, as in Example 1; and proceed to find the indicated probability.

b. In a 118-year period, how many years are expected to have 10 hurricanes?

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