Problem 51 - Basic Statistics for Business & Economics

Question

Recent crime reports indicate that 3.1 motor vehicle thefts occur each minute in the United States. Assume that the distribution of thefts per minute can be approximated by the Poisson probability distribution. a. Calculate the probability exactly four thefts occur in a minute. b. What is the probability there are no thefts in a minute? c. What is the probability there is at least one theft in a minute?

Step-by-Step Solution

Answer

a. 0.1680; b. 0.0450; c. 0.9550.

Step 1: Understand the Poisson Distribution

The Poisson distribution is used to model the number of events that occur within a fixed interval of time or space. The number of events is specified by \( \lambda \), the average rate of occurrence. Here, \( \lambda = 3.1 \) thefts per minute.

Step 2: Calculate Probability of Exactly Four Thefts (Question a)

The probability of observing \( k \) events in an interval is given by the Poisson probability formula: \( P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} \). For exactly four thefts, \( k = 4 \): \[P(X = 4) = \frac{3.1^4 \cdot e^{-3.1}}{4!} \approx \frac{92.3521 \cdot 0.045049202}{24} \approx 0.168031355\]Therefore, the probability of exactly four thefts occurring in a minute is approximately 0.1680.

Step 3: Calculate Probability of No Thefts (Question b)

For zero thefts, \( k = 0 \), the probability formula becomes:\[P(X = 0) = \frac{3.1^0 \cdot e^{-3.1}}{0!} = e^{-3.1} \approx 0.045049202\]Hence, the probability of no thefts occurring in a minute is about 0.0450.

Step 4: Calculate Probability of At Least One Theft (Question c)

The probability of at least one theft occurring can be found by subtracting the probability of no thefts from 1:\[P(X \geq 1) = 1 - P(X = 0) = 1 - 0.045049202 \approx 0.954950798\]Thus, the probability of at least one theft occurring in a minute is approximately 0.9550.

Key Concepts

Probability

Probability helps us understand the likelihood of certain events happening within a specific context. In statistics, particularly with the Poisson distribution, probability is about assessing how likely it is for a number of events, such as thefts, to occur during a given time period. The Poisson distribution is a probabilistic model that provides a method to calculate this likelihood based on an average rate of occurrence.

Using the Poisson formula, one can calculate the probability of exactly or at least a specific number of events happening. This requires knowing the average rate (\( \lambda \)), which informs how frequent these events happen on average over a set time. For example, the probability of exactly four motor vehicle thefts in one minute can be found by applying the formula, resulting in a calculated probability of 0.1680.

Thefts

Thefts, specifically motor vehicle thefts, are treated as random events in the context of probability distribution theory. When we analyze the occurrence of thefts over time using statistics, the Poisson distribution is useful to model this behavior.

Given that on average, there are 3.1 thefts per minute, this parameter becomes crucial in calculating the likelihood of varying theft outcomes. By understanding the rate and applying probability models, it becomes possible to derive meaningful insights about crime patterns, which are important for resource allocation by law enforcement and for public awareness.

Whether predicting the frequency of thefts precisely helps in planning safety measures or informing insurance policies, recognizing the statistical patterns of thefts facilitates better preparation and response strategies.

Average Rate

The average rate, denoted as \( \lambda \), is a key parameter in the Poisson distribution formula. It represents the average number of occurrences for an event within a given time frame. For motor vehicle thefts, \( \lambda = 3.1 \), which implies that on average, there are 3.1 thefts per minute.

Understanding the average rate is essential for predicting the probability of different outcomes. This is because it captures the historical frequency of the events and serves as a baseline for calculations. For instance, this average informs predictions such as the probability of zero thefts (calculated to be approximately 0.0450) or at least one theft in a minute.

Setting this average rate in the Poisson formula allows us to explore other probability scenarios and make informed decisions based on statistical expectations.

Events in an Interval

Events in an interval refer to occurrences happening within a specific time frame. In statistical probability, modeling these events using the Poisson distribution helps estimate how many will likely occur. This involves determining the probability of various numbers of events, such as thefts, and requires knowing both the average rate of occurrence and the interval duration.

For instance, if the interval is one minute and we're interested in thefts, knowing that there is an average of 3.1 thefts per minute allows us to model potential outcomes accurately. We can assess scenarios such as having no thefts or multiple thefts by substituting values into the Poisson probability formula.

This methodology aids in understanding not just the likelihood of single events but also provides insight into the distribution of outcomes over time, which is useful in practical applications like urban planning or emergency preparedness.