Problem 9 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The number of accidents that occur at a certain intersection known as "Five Corners" on a Friday afternoon between the hours of 3 p.m. and 6 p.m., along with the corresponding probabilities, are shown in the following table. Find the expected number of accidents during the period in question. $$\begin{array}{lccccc}\hline \text { Accidents } & 0 & 1 & 2 & 3 & 4 \\ \hline \text { Probability } & .935 & .030 & .020 & .010 & .005 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The expected number of accidents on a Friday afternoon between 3 p.m. and 6 p.m. at the "Five Corners" intersection is \(0.120\).

Step 1: Identify the corresponding probabilities for each accident

We have the following probabilities and their associated number of accidents in the table: Accidents: 0 | 1 | 2 | 3 | 4 ------------------------------------------------ Probabilities: 0.935 | 0.030 | 0.020 | 0.010 | 0.005

Step 2: Calculate the product of accidents and their probabilities

Multiply each accident value by its corresponding probability: 0 × 0.935 = 0 1 × 0.030 = 0.030 2 × 0.020 = 0.040 3 × 0.010 = 0.030 4 × 0.005 = 0.020

Step 3: Sum up the products to find the expected value

Add the products found in Step 2: Expected value (E) = 0 + 0.030 + 0.040 + 0.030 + 0.020 = 0.120 The expected number of accidents on a Friday afternoon between 3 p.m. and 6 p.m. at the "Five Corners" intersection is 0.120.

Key Concepts

Probability Distribution

A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. In our exercise, we are examining the number of accidents at a busy intersection during a specific time period. For probability distributions with discrete outcomes, like accidents in our case, each potential outcome has an associated probability. This list of probabilities needs to sum up to 1, which indicates all possible outcomes together represent the whole scenario.

• In our example, the intersection can experience anywhere from 0 to 4 accidents during the specified time frame.
• Each number of accidents has a probability assigned to it, i.e., 0 accidents with a 0.935 probability, and so on up to 4 accidents with a 0.005 probability.

Understanding how these probabilities relate to potential outcomes is crucial for calculating the expected value, which provides an average number of accidents based on these probabilities.

Discrete Random Variables

Discrete random variables are variables that can take on a countable number of distinct values. In our context, the random variable is the number of accidents during a specific time period at the intersection.

• These variables differ from continuous random variables, which can take on an infinite number of values within a given range.
• For discrete random variables, each specific value has a probability attached to it. This makes them easier to manage since we can directly calculate probabilities for specific outcomes.

To find the expected value of a discrete random variable, multiply each outcome by its probability, and then sum up these products. For example, in our exercise, the expected number of accidents was calculated by combining each accident count with its probability, then summing all these results to obtain the expected value.

Statistical Analysis

Statistical analysis involves methods for collecting, reviewing, and drawing conclusions from data. One application is finding the expected value, as shown in our exercise with accident occurrences.

• This calculation helps in understanding the data by summarizing it into a single metric that offers insights into what to generally expect.
• Beyond simple calculations, statistical analysis often involves interpreting results to guide decision-making or predict future events.

In the example provided, statistical analysis was employed to calculate the expected number of accidents, equipping authorities or planners with practical information needed to make informed decisions about traffic management or other preventive measures. The expected value, though a single number, provides a comprehensive understanding of the pattern and likelihood of various accident scenarios.