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Conditional Probability: Hail Damage In western Kansas, the summer density of hailstorms is estimated at about \(2.1\) storms per 5 square miles. In most cases, a hailstorm damages only a relatively small area in a square mile (Reference: Agricultural Statistics, U.S. Department of Agriculture). A crop insurance company has insured a tract of 8 square miles of Kansas wheat land against hail damage. Let \(r\) be a random variable that represents the number of hailstorms this summer in the 8 -square-mile tract. (a) Explain why a Poisson probability distribution is appropriate for \(r\). What is \(\lambda\) for the 8 -square-mile tract of land? Round \(\lambda\) to the nearest tenth so that you can use Table 4 of Appendix II for Poisson probabilities. (b) If there already have been two hailstorms this summer, what is the probability that there will be a total of four or more hailstorms in this tract of land? Compute \(P(r \geq 4 \mid r \geq 2)\). (c) If there already have been three hailstorms this summer, what is the probability that there will be a total of fewer than six hailstorms? Compute \(P(r<6 \mid r \geq 3)\).

Step by step solution

01

Understanding Poisson Distribution Applicability

A Poisson distribution is used for counting the number of events (such as hailstorms) occurring within a fixed interval of time or space, given that these events occur independently of each other. The randomness, independence, and the relatively low probability of these events per unit area make the Poisson distribution appropriate.

02

Calculating Lambda for the Poisson Distribution

The rate of storms is given as 2.1 storms per 5 square miles. For an 8 square mile area, the average number of storms (\lambda) would be calculated as:\[\lambda = \frac{2.1}{5} \times 8 = 3.36\]Rounding \lambda to the nearest tenth gives \lambda = 3.4.

03

Finding Probability of Four or More Hailstorms Given Two Already Occurred

To find \( P(r \geq 4 \mid r \geq 2) \), we use the formula: \[P(A \mid B) = \frac{P(A \cap B)}{P(B)}\] Here, \( A \) is the event of having four or more storms, and \( B \) is the event of having at least two storms. First, find \( P(r \geq 4) \) using the complement rule:\[P(r \geq 4) = 1 - P(r < 4) = 1 - (P(r = 0) + P(r = 1) + P(r = 2) + P(r = 3))\]Then, find \( P(r \geq 2) \), and use the formula above.

04

Calculating P(r

Apply the Poisson formula:\[P(r = k) = \frac{e^{-\lambda} \lambda^k}{k!}\]Calculate for each value \( k = 0, 1, 2, 3 \), then sum these probabilities to obtain \( P(r < 4) \). Given \( \lambda = 3.4 \), use Appendix II tables if needed for greater accuracy.

05

Probability of Fewer than Six Given Three Hailstorms

For \( P(r<6 \mid r \geq 3) \), use the conditional probability formula. First, find \( P(r<6 \cap r \geq 3) = P(3 \leq r < 6) \) and \( P(r \geq 3) \), then apply:\[P(A \mid B) = \frac{P(A \cap B)}{P(B)}\]Compute each probability value using the Poisson distribution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Poisson Distribution

The Poisson distribution is a probability distribution that models how often an event will happen over a certain period of time or space. It's particularly useful for counting occurrences when these events happen independently. For example, hailstorms over a wheat land are events that can be anticipated using this distribution because they are random and independent of other hailstorms in nearby areas.

• The Poisson distribution is characterized by the parameter \( \lambda \), which is the average number of occurrences in the fixed interval.
• In the case of our hailstorm example, \( \lambda \) is calculated based on the average rate of storms in a specified area.

Understanding this concept is key as it helps in predicting events such as the number of storms. This distribution is most applicable when dealing with rare events over large scales or for particular time periods.

Random Variable

A random variable is a numerical representation of the outcomes of a random phenomenon. In essence, it is a variable whose possible values are numerical outcomes of a random event. For our hailstorm scenario, the random variable \( r \) represents the number of hailstorms occurring in the specified area of 8 square miles during the summer.
Random variables come in two types: discrete and continuous. A discrete random variable, like \( r \), counts things, such as the number of occurrences of an event.

• \( r \) is a discrete random variable as it counts the number of storm events.
• Each different possible count (e.g., 0, 1, 2 hailstorms) has a probability associated with it.
• Understanding how these probabilities are derived and interpreted is essential for predicting and making decisions based on these variables.

Probability Calculation

Calculating probability involves determining the likelihood of a specific event occurring. For hailstorms, we want to understand how probable it is for a certain number of hailstorms to occur given an average rate. Specifically, we are interested in probabilities like \( P(r \geq 4 \mid r \geq 2) \) or \( P(r < 6 \mid r \geq 3) \).

• These calculations often use formulas and tables derived from statistical theories such as the Poisson distribution.
• Probability calculations may also use techniques such as the complement rule, which helps in determining probabilities by considering the opposite of the desired event.

Understanding probability calculations can offer insights into expected outcomes and assist in risk management, such as determining insurance premiums.

Conditional Probability Formula

The conditional probability formula is used to find the probability of an event occurring given that another event has already occurred. It expresses how the probability is restricted to a particular condition. For example, if we know it has already hailed twice, we can use the formula to find the probability of four or more hailstorms by the end of the summer:
\[P(A \mid B) = \frac{P(A \cap B)}{P(B)}\]

• Here, \( A \) represents the event for which we want to find the probability, and \( B \) is the condition we already know to be true.
• \( P(A \cap B) \) is the probability that both events \( A \) and \( B \) occur.
• \( P(B) \) is the probability that condition \( B \) occurs.

This formula is invaluable in predicting outcomes under known conditions, offering a refined probability that accounts for prior occurrences.