Problem 19 - Algebra Form and Function

Question

Table 17.19 gives the vehicle occupancy for people driving to work in 1990 as determined by the US Census. For instance, 84,215,000 people drove alone and 12,078,000 people drove in 2 -person car pools. Picking at random, what is the probability, given as a percentage, that: (a) A commuter drives to work alone? (b) A vehicle carries 4 or more people?$$ \begin{array}{c|c|c|c|c|c|c|c} \hline \text { Occupancy } & 1 & 2 & 3 & 4 & 5 & 6 & 7+ \\ \hline \text { People, } 1000 \mathrm{~s} & 84,215 & 12,078 & 2,001 & 702 & 209 & 97 & 290 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Answer: Based on the given data, the probability that a commuter drives to work alone is 84.59%, and the probability that a vehicle carries 4 or more people is 1.30%.

Step 1: Calculate the total number of commuters

First, we will sum up all the numbers in the given table to find the total number of people driving to work. This will be used as the denominator while calculating probabilities. Total number of commuters = 1-person occupancy + 2-person occupancy + 3-person occupancy + 4-person occupancy + 5-person occupancy + 6-person occupancy + 7+ person occupancy \(Total = 84215 + 12078 + 2001 + 702 + 209 + 97 + 290 = 99,592\) (in thousands)

Step 2: Calculate the probability that a commuter drives to work alone

Now, we will divide the number of people driving alone by the total number of people driving to work. \(P(\textrm{driving alone}) = \frac{\textrm{1-person occupancy}}{\textrm{Total number of commuters}}\) \(P(\textrm{driving alone}) = \frac{84215}{99592} = 0.8459\) Now, convert the probability to percentage: \(0.8459 * 100 = 84.59\%\) Thus, the probability that a commuter drives to work alone is 84.59%.

Step 3: Calculate the probability that a vehicle carries 4 or more people

We need to find the total number of people in vehicles with 4 or more occupants and divide this by the total number of people driving to work to calculate this probability. Number of 4+ person occupancy = 4-person occupancy + 5-person occupancy + 6-person occupancy + 7+ person occupancy \(4+ \textrm{ occupancy} = 702 + 209 + 97 + 290 = 1298\) (in thousands) \(P(\textrm{4 or more occupants}) = \frac{\textrm{4+ person occupancy}}{\textrm{Total number of commuters}}\) \(P(\textrm{4 or more occupants}) = \frac{1298}{99592} = 0.0130\) Now, convert the probability to percentage: \(0.0130 * 100 = 1.30\%\) Thus, the probability that a vehicle carries 4 or more people is 1.30%.

Key Concepts

Vehicle Occupancy

Vehicle occupancy refers to the number of passengers that a single vehicle carries. It plays a significant role in understanding commuting behaviors and traffic patterns. In our context, vehicle occupancy is categorized by the number of people within a vehicle as they commute to work.
Understanding this concept helps us analyze and optimize transportation systems, evaluate environmental impact, and improve road usage efficiency. In our exercise, we have data on how many people drove alone, in two-person pools, and so on, until vehicles that carry seven or more people. This data allows us to calculate commuting patterns and the likelihood of these occurrences. Knowing vehicle occupancy can also help in making predictions about traffic congestion and in planning infrastructure requirements.

US Census Data

US Census data offers comprehensive statistical information about the population and its habits. Conducted every ten years, it helps in gathering data about various facets of citizens' lives, including commuting patterns. This makes it an invaluable source for understanding nationwide trends such as vehicle occupancy in commuting.

In this exercise, the US Census data from 1990 provides numbers of people commuting in each vehicle occupancy category. Such data not only reflects commuting preferences of that era but also serves as a comparative basis for observing changes over time. Analysts can use this information to understand shifts in commuting habits and their implications on economic and urban planning.

Commuting Patterns

Commuting patterns describe how people travel from their homes to their places of work and back. These patterns are shaped by various factors, including geography, urban design, personal preferences, and economic conditions.
By analyzing commuting patterns, researchers and policymakers can identify the efficiency of public transit systems, the prevalence of carpooling, and the reliance on personal vehicles. In our scenario, the commuting pattern data from the 1990 census helps us understand how likely individuals were to carpool or travel alone to work. Understanding these patterns aids in making informed decisions regarding urban development, public transport systems, and even environmental policies.

Percentage Calculation

Calculating percentages is crucial in interpreting probability data. It provides an easily understood figure that conveys the likelihood of an event occurring as part of a whole.
To calculate a percentage, you can use the formula:

• Identify the part you want to express as a percentage.
• Divide the part by the whole (total number of occurrences or possibilities).
• Multiply the result by 100.

In this problem, we use percentage calculations to determine how likely it is for commuters to drive alone or be in vehicles with four or more occupants. Calculating these probabilities in percentages gives an intuitive understanding of these commuting choices. It allows us to easily communicate these findings to others, making data-driven insights more accessible.