Skip to main content
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 5.3.7b
Chapter 5, Problem 5.3.7b

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 no hurricanes?

Verified step by step guidance
1
Understand the problem: The Poisson distribution is used to model the number of events (hurricanes) occurring in a fixed interval of time (years). The mean number of hurricanes per year is given as 5.5. We are tasked with finding the expected number of years with no hurricanes over a 118-year period.
Step 1: Recall the formula for the Poisson probability mass function (PMF): P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the mean number of events, k is the number of events, and e is the base of the natural logarithm (approximately 2.718).
Step 2: For this problem, we are interested in the probability of having no hurricanes in a single year (k = 0). Substitute k = 0 and λ = 5.5 into the PMF formula: P(X = 0) = (5.5^0 * e^(-5.5)) / 0!. Simplify this expression, noting that 0! = 1 and 5.5^0 = 1.
Step 3: Once you calculate P(X = 0), interpret it as the probability of having no hurricanes in a single year. To find the expected number of years with no hurricanes over a 118-year period, multiply this probability by the total number of years: Expected years = P(X = 0) * 118.
Step 4: Perform the calculations step by step to determine the final expected number of years with no hurricanes. Ensure that you use accurate values for e^(-5.5) and complete the multiplication with 118 to arrive at the result.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

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 natural disasters, where the events are independent of each other.
Recommended video:
Guided course
06:38
Intro to Frequency Distributions

Expected Value

The expected value is a key concept in probability that represents the average outcome of a random variable over a large number of trials. In the context of the Poisson distribution, the expected number of occurrences can be calculated by multiplying the average rate (mean) by the number of intervals considered, providing a basis for predicting outcomes.
Recommended video:
Guided course
04:14
Expected Value (Mean) of Random Variables

Probability of Zero Events

In a Poisson distribution, the probability of observing zero events in a given interval can be calculated using the formula P(X=0) = e^(-λ), where λ is the mean number of events. This concept is crucial for determining how many years in a specified period are expected to have no hurricanes, as it directly relates to the average rate of occurrence.
Recommended video:
05:54
Probability of Multiple Independent Events
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.

Textbook Question

Lottery. In Exercises 15–20, refer to the accompanying table, which describes probabilities for the California Daily 4 lottery. The player selects four digits with repetition allowed, and the random variable x is the number of digits that match those in the same order that they are drawn (for a “straight” bet).


Using Probabilities for Significant Events


a. Find the probability of getting exactly 2 matches.

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


a. Find the probability that in a year, there will be 7 hurricanes.

1
views
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?

Textbook Question

In Exercises 31 and 32, assume that hybridization experiments are conducted with peas having the property that for offspring, there is a 0.75 probability that a pea has green pods (as in one of Mendel’s famous experiments).


Hybrids Assume that offspring peas are randomly selected in groups of 16.


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

1
views
Textbook Question

One of Mendel’s famous experiments with peas resulted in 580 offspring, and 152 of them were yellow peas. Mendel claimed that under the same conditions, 25% of offspring peas would be yellow. Assume that Mendel’s claim of 25% is true, and assume that a sample consists of 580 offspring peas.


b. Find the probability of exactly 152 yellow peas.


1
views