5. Binomial Distribution & Discrete Random Variables

Binomial Distribution

5. Binomial Distribution & Discrete Random Variables

Binomial Distribution

1:50 minutes

Problem 5a

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.

Verified step by step guidance

Step 1: Understand the Poisson distribution. The Poisson distribution is used to model the probability of a given number of events occurring in a fixed interval of time or space, given a known average rate of occurrence (mean). The formula for the Poisson probability is P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the mean number of occurrences, k is the number of occurrences, and e is the mathematical constant approximately equal to 2.718.

Step 2: Identify the given values. From the problem, the mean number of hurricanes per year (λ) is 5.5, and we are tasked with finding the probability of exactly 7 hurricanes (k = 7).

Step 3: Substitute the values into the Poisson formula. Replace λ with 5.5 and k with 7 in the formula: P(X = 7) = (5.5^7 * e^(-5.5)) / 7!.

Step 4: Break down the calculation. Compute each component step-by-step: (1) Calculate 5.5^7, (2) Compute e^(-5.5), (3) Find the factorial of 7 (7!), and (4) Divide the product of 5.5^7 and e^(-5.5) by 7!.

Step 5: Interpret the result. Once the probability is calculated, it represents the likelihood of observing exactly 7 hurricanes in a year under the given conditions. This value can be compared to other probabilities to understand the distribution of hurricane occurrences.

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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 occur independently. The formula for the Poisson probability mass function is P(X=k) = (λ^k * e^(-λ)) / k!, where λ is the average rate, k is the number of occurrences, and e is Euler's number.

Mean (λ)

In the context of the Poisson distribution, the mean (denoted as λ) represents the average number of occurrences of the event in a specified interval. For this question, λ is given as 5.5, indicating that, on average, there are 5.5 hurricanes per year in the United States. This parameter is crucial for calculating the probabilities of different outcomes using the Poisson formula.

Probability Calculation

To find the probability of a specific number of events occurring in a Poisson distribution, one uses the Poisson probability mass function. For this question, to find the probability of exactly 7 hurricanes in a year, you would substitute k=7 and λ=5.5 into the formula. This calculation will yield the likelihood of observing that exact number of hurricanes, which is essential for understanding the variability and risk associated with hurricane occurrences.

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Textbook Question

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c. Find the probability of 40 or more first lines for Democrats.

Textbook Question

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For groups of five drivers, find the mean and standard deviation for the numbers of drivers who say that they text while driving.

Textbook Question

Internet Traffic Data Set 27 “Internet Traffic” includes 9000 arrivals of Internet traffic at the Digital Equipment Corporation, and those 9000 arrivals occurred over a period of 19,130 thousandths of a minute. Let the random variable x represent the number of such Internet traffic arrivals in one thousandth of a minute. It appears that these Internet arrivals have a Poisson distribution. If we want to use Formula 5-9 to find the probability of exactly 2 arrivals in one thousandth of a minute, what are the values of μ, x, and e that would be used in that formula?

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

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

Hurricanes

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

Show more practices

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

Key Concepts

Poisson Distribution

Mean (λ)

Probability Calculation

Watch next

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.