Problem 17 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The number of cars entering a tunnel leading to an airport in a major city over a period of 200 peak hours was observed, and the following data were obtained: $$ \begin{array}{rc} \hline \begin{array}{l} \text { Number of } \\ \text { Cars, } x \end{array} & \begin{array}{c} \text { Frequency of } \\ \text { Occurrence } \end{array} \\ \hline 01000 & 15 \\ \hline \end{array} $$ a. Describe an appropriate sample space for this experiment. b. Find the empirical probability distribution for this experiment.

Step-by-Step Solution

Answer

a. The sample space S for this experiment is: S = {0 < x ≤ 200, 200 < x ≤ 400, 400 < x ≤ 600, 600 < x ≤ 800, 800 < x ≤ 1000, x > 1000} b. The empirical probability distribution for this experiment is: P(x) = {0.075 (0 < x ≤ 200), 0.1 (200 < x ≤ 400), 0.175 (400 < x ≤ 600), 0.35 (600 < x ≤ 800), 0.225(800 < x ≤ 1000), 0.075 (x > 1000)}

Step 1: Sample Space

The sample space is the set of all possible outcomes for this experiment. In this case, it refers to the possible numbers of cars entering the tunnel during peak hours. From the data, we know that the range is from 0 to more than 1000 cars. Thus, the sample space S for this experiment can be represented as: S = {0 < x ≤ 200, 200 < x ≤ 400, 400 < x ≤ 600, 600 < x ≤ 800, 800 < x ≤ 1000, x > 1000} #b. Find the empirical probability distribution for this experiment.#

Step 2: Total Frequency

We will first find the total frequency (N) by adding up all the frequencies in the given data: N = 15 + 20 + 35 + 70 + 45 + 15 = 200

Step 3: Empirical Probability Distribution Calculation

Now we will calculate the empirical probability for each category in the sample space. P(0 < x ≤ 200) = frequency of (0 < x ≤ 200) ÷ N = 15 ÷ 200 = 0.075 P(200 < x ≤ 400) = frequency of (200 < x ≤ 400) ÷ N = 20 ÷ 200 = 0.1 P(400 < x ≤ 600) = frequency of (400 < x ≤ 600) ÷ N = 35 ÷ 200 = 0.175 P(600 < x ≤ 800) = frequency of (600 < x ≤ 800) ÷ N = 70 ÷ 200 = 0.35 P(800 < x ≤ 1000) = frequency of (800 < x ≤ 1000) ÷ N = 45 ÷ 200 = 0.225 P(x > 1000) = frequency of (x > 1000) ÷ N = 15 ÷ 200 = 0.075

Step 4: Empirical Probability Distribution

Based on the calculations, we can write the empirical probability distribution for this experiment as: P(x) = {0.075 (0 < x ≤ 200), 0.1 (200 < x ≤ 400), 0.175 (400 < x ≤ 600), 0.35 (600 < x ≤ 800), 0.225(800 < x ≤ 1000), 0.075 (x > 1000)}

Key Concepts

Understanding Sample Space

When tackling any probability problem, the concept of a sample space is fundamental. It's the foundation upon which all probability is built. The sample space of an experiment, often denoted as 'S', includes every single possible outcome that could occur. In our exercise, the sample space concerns the number of cars entering a tunnel over a given period - a discrete set of intervals representing vehicle counts.

To properly determine the sample space, one must consider all categories inclusive of the observed data. It's like picturing a big net that can catch all the possible events. This set for the car counts ranges from 0 to greater than 1000, divided into intervals. A well-constructed sample space allows us to perform accurate probability calculations. It is the bedrock of understanding empirical probability distributions and helps ensure that we account for all potential outcomes without leaving anything to chance.

Decoding Frequency Distribution

A frequency distribution is like a snapshot of data that helps us see patterns. Think of it as a tally chart where you count how often each outcome happens. This distribution is visually similar to a bar graph where the height of each bar represents the frequency, or count, of occurrences for each outcome. In our exercise, the frequency distribution tells us how often a specific range of cars enter the tunnel.

For instance, we count that on 15 occasions, the number of cars was between 0 and 200, and on 20 occasions, it was between 200 and 400, and so on. These frequencies help us measure the relative likelihood of different outcomes within the sample space. When dealing with real data like the car counts, a frequency distribution provides a helpful overview, which is a step towards finding what's typical or unusual in the pattern of data. For students, it's crucial to understand that the heights of each 'bar' in the distribution play a direct role in determining the probabilities of each outcome.

Probability Calculation Essentials

The probability calculation is the mathematical determination of how likely an event is to occur. It's like a bet on the future but one that's based on past information. The empirical probability is specifically derived from the frequency of past occurrences, hence it is grounded in actual experience or observation. To calculate the empirical probability, simply divide the frequency of a particular outcome by the total number of trials (observations).

In our example, we calculated the probability for each car count interval by dividing its frequency by the total frequency (200 in this case). This gives us the likelihood, expressed as a decimal, for each interval. The sum of all these probabilities should be 1, since it's certain that one of the outcomes in the sample space will happen. For students actually calculating these probabilities, remember: it's a way to quantify uncertainty based on what has been observed and is a key concept in the practice of statistics.