Problem 51 - Basic Statistics for Business and Economics with Student

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. 0.168, b. 0.045, c. 0.955

Step 1: Define the Problem

We're given that the average number of motor vehicle thefts per minute, \( \lambda \), is 3.1. We need to calculate probabilities using the Poisson distribution for different scenarios.

Step 2: Understanding the Poisson Probability Formula

The Poisson probability formula is \( P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!} \), where \( e \approx 2.71828 \), \( \lambda \) is the average rate (3.1 in this case), and \( k \) is the number of occurrences.

Step 3: Calculate Probability of Exactly Four Thefts

Use the Poisson formula with \( k = 4 \). \[ P(X = 4) = \frac{e^{-3.1} \times 3.1^4}{4!} \].Calculate each part: - \( e^{-3.1} \approx 0.0450492 \) - \( 3.1^4 = 92.3521 \) - \( 4! = 24 \).Thus, \[ P(X = 4) = \frac{0.0450492 \times 92.3521}{24} \approx 0.1680316 \].

Step 4: Calculate Probability of No Thefts

Use the Poisson formula with \( k = 0 \). \[ P(X = 0) = \frac{e^{-3.1} \times 3.1^0}{0!} = e^{-3.1} \approx 0.0450492 \].

Step 5: Calculate Probability of At Least One Theft

The probability of at least one theft is the complement of having no thefts. \[ P(X \geq 1) = 1 - P(X = 0) = 1 - 0.0450492 \approx 0.9549508 \].

Key Concepts

Probability Calculation

A probability calculation involves determining how likely an event is to occur given certain conditions. In the context of the Poisson distribution, it's essential to first identify the average rate at which an event happens, denoted as \( \lambda \). For example, in cases of motor vehicle thefts, if \( \lambda = 3.1 \), it indicates that, on average, 3.1 thefts happen every minute.

For calculations with the Poisson distribution, we utilize the formula:

• \( P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!} \)

Here, \( e \) represents Euler's number (approximately 2.71828).\( k \) stands for the number of occurrences we want the probability for, like exactly four thefts in a minute.

Breaking down the formula demonstrates how each variable contributes to the calculation. The exponential component \( e^{-\lambda} \) shows the influence of the time frame while \( \lambda^k \) connects the average rate with the exact number of events (k). Factorial \( k! \) adjusts for the multiple ways the events can occur.

Probability Distribution

Probability distribution is a statistical function that describes all the possible values and likelihoods that a random variable can take within a given range. The Poisson distribution, in particular, is ideal for events that occur independently over a fixed period.

A unique feature of the Poisson distribution is its focus on counts of events, like the count of car thefts clustered around the average \( \lambda \). It assumes that in a very short period, the likelihood of more than one event occurring is negligible. This makes it particularly useful for scenarios like motor vehicle thefts reported to occur at random times.

For example, using \( \lambda = 3.1 \), we can calculate the probability of different scenarios, such as exactly 0 thefts, exactly 4 thefts, or at least 1 theft in a minute. This distribution allows us to draw insights into how frequently certain numbers of events are likely to occur and plan resources accordingly.

Motor Vehicle Thefts

Understanding the statistics and probabilities behind motor vehicle thefts can guide law enforcement and policy decisions. If crime reports estimate that about 3.1 thefts occur every minute in the US, it signifies a serious and persistent challenge that community safety measures must address.

By applying the Poisson distribution to this data, we can derive probabilities for various scenarios, such as the chance of exact numbers of thefts occurring in any given minute. For instance:

• The probability of exactly four thefts helps assess the regularity of such incidences.
• Calculating the chances of no thefts, which is quite rare, provides insights into the effectiveness of deterrents.
• Understanding the likelihood of at least one theft occurring highlights the urgency for ongoing vigilance and resource allocation.

The insights gained from these probability calculations play a crucial role in risk assessment and prevention strategies.