5. Binomial Distribution & Discrete Random Variables

Binomial Distribution

5. Binomial Distribution & Discrete Random Variables

Binomial Distribution

2:06 minutes

Problem 6a

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 no hurricanes.

Verified step by step guidance

Step 1: Recall the formula for the Poisson probability distribution: P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the mean number of occurrences, k is the number of occurrences we are interested in, and e is the base of the natural logarithm (approximately 2.718).

Step 2: Identify the given values from the problem. Here, λ = 5.5 (the mean number of hurricanes per year) and k = 0 (since we are finding the probability of no hurricanes in a year).

Step 3: Substitute the values into the formula. This gives P(X = 0) = (5.5^0 * e^(-5.5)) / 0!. Note that any number raised to the power of 0 is 1, and 0! (zero factorial) is also equal to 1.

Step 4: Simplify the expression. The numerator becomes 1 (since 5.5^0 = 1), and the denominator is 1 (since 0! = 1). This leaves P(X = 0) = e^(-5.5).

Step 5: To find the final probability, calculate e^(-5.5) using a calculator or software. This will give the probability of having no hurricanes in a 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 the number of hurricanes in a year, where the events are independent of each other.

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 this question, λ is given as 5.5, indicating that, on average, there are 5.5 hurricanes per year in the United States.

Calculating Probability with Poisson

To find the probability of observing a specific number of events in a Poisson distribution, the formula P(X=k) = (e^(-λ) * λ^k) / k! is used, where P(X=k) is the probability of k events occurring, e is the base of the natural logarithm, and k! is the factorial of k. For this question, to find the probability of zero hurricanes, k would be 0.

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Textbook Question

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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

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Textbook Question

Hurricanes

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Textbook Question

Hurricanes

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Textbook Question

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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 no hurricanes.

Key Concepts

Poisson Distribution

Mean (λ) in Poisson Distribution

Calculating Probability with Poisson

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