Table of contents

1. Intro to Stats and Collecting Data1h 14m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically2h 5m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables3h 6m
6. Normal Distribution and Continuous Random Variables2h 11m
7. Sampling Distributions & Confidence Intervals: Mean3h 23m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample3h 29m
10. Hypothesis Testing for Two Samples4h 50m
11. Correlation1h 6m
12. Regression1h 50m
13. Chi-Square Tests & Goodness of Fit1h 57m
14. ANOVA1h 57m

5. Binomial Distribution & Discrete Random Variables

Binomial Distribution

Statistics

5. Binomial Distribution & Discrete Random Variables

Binomial Distribution

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2:29 minutes

Problem 6b

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 no hurricanes?

Verified step by step guidance

Step 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 λ = 5.5. We are tasked with finding the expected number of years with no hurricanes over a 118-year period.

Step 2: Recall the formula for the Poisson probability mass function (PMF): P(X = k) = (λ^k * e^(-λ)) / k!, where X is the number of events, λ is the mean number of events, k is the specific number of events (in this case, k = 0 for no hurricanes), and e is the base of the natural logarithm.

Step 3: Calculate the probability of having no hurricanes in a single year using the Poisson PMF. Substitute λ = 5.5 and k = 0 into the formula: P(X = 0) = (5.5^0 * e^(-5.5)) / 0!. Simplify the expression, noting that 0! = 1 and 5.5^0 = 1.

Step 4: Once the probability of no hurricanes in a single year is determined, multiply this probability by the total number of years (118) to find the expected number of years with no hurricanes. Use the formula: Expected years = P(X = 0) * 118.

Step 5: Interpret the result. The final value represents the expected number of years out of 118 that will have no hurricanes, based on the given mean of 5.5 hurricanes per year.

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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 natural disasters, where the events are independent of each other.

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.

Probability of No 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, based on the average rate.

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Related Practice

Textbook Question

For the distribution described in Exercise 1, find the probability of exactly 2 arrivals in one thousandth of a minute.

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.

Textbook Question

Hurricanes

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

Textbook Question

Hurricanes

a. Find the probability that in a year, there will be no hurricanes.

Textbook Question

Graphical Analysis In Exercises 3–5, the histogram represents a binomial distribution with five trials. Match the histogram with the appropriate probability of success p. Explain your reasoning.

a. p = 0.25

b. p = 0.50

c. p = 0.75

Textbook Question

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Textbook Question

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Textbook Question

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