Problem 24 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The area of a histogram associated with a probability distribution is a number between 0 and 1 .

Step-by-Step Solution

Answer

The statement is true. The area of a histogram associated with a probability distribution is a number between 0 and 1 because each bar in the histogram represents the probability of a specific outcome, and the probability of each outcome must be between 0 and 1 according to the properties of a probability distribution.

Step 1: Recall the properties of a probability distribution

A probability distribution must satisfy two properties: 1. The probability of each outcome is between 0 and 1, inclusive: \(0 \le P(x) \le 1\). 2. The sum of probabilities of all outcomes is equal to 1: \(\sum P(x) = 1\).

Step 2: Analyzing the area of a histogram

In a histogram, the area of any particular bar corresponds to the probability of that particular outcome. Since a histogram represents the probability distribution, it must also follow the properties of a probability distribution, as mentioned above.

Step 3: Answer the question

The area of a histogram associated with a probability distribution is a number between 0 and 1 because each bar in the histogram represents the probability of a specific outcome, and the probability of each outcome must be between 0 and 1 according to the properties of a probability distribution. The statement in the exercise is true.

Key Concepts

Histogram Area and Probability

A histogram is a graphical representation often used to showcase the frequency of data points within certain ranges, or 'bins'. In the context of probability, histograms visualize the probability distribution of a random variable. Importantly, the area of each rectangle or 'bar' of a histogram corresponds to the probability of that variable's outcome falling within the specific range the bar covers. With this knowledge, one can examine why the area of a histogram associated with a probability distribution has considerable importance.

To understand the relationship, consider that each bar's height is determined by the probability of the outcome, and the width is set at a standard value (usually 1). So, if the height represents probability and the width represents a single count, the area (height times width) will naturally represent the probability as well, considering the width is constant. The total area under the histogram will add up to 1, which aligns with the fundamental principle that the sum of probabilities of all outcomes in a probability distribution equals 1.

Properties of Probability

The properties of probability are essential rules that govern how probabilities are assigned to events in a probability distribution. The first property states that the probability of any outcome is a non-negative number that does not exceed 1, formally written as \( 0 \le P(x) \le 1 \). This is intuitive as a probability is essentially the measure of certainty that a particular outcome will occur, and therefore, it cannot be less than 0% or more than 100% — which are represented by the values 0 and 1, respectively.

The second crucial property highlights that probabilities in a distribution must sum to 1. This is akin to saying that if we consider all possible outcomes of a random experiment, there is a certainty (100%) that one of the outcomes will indeed occur. It's the foundation for the creation of probability models and is essential for the evaluation of events with certain and uncertain outcomes. These properties ensure that probability distributions are meaningful and that the probabilities are coherent.

Sum of Probabilities

Diving deeper into the sum of probabilities, it's a foundational concept that the total probability of all possible outcomes of a probability distribution must equal 1. This idea is called the normalization condition of probabilities and is represented mathematically as \( \sum P(x) = 1 \), where the sum is over all possible outcomes 'x'.

This principle allows us to make sense of the relative likelihood of various outcomes. For example, if you flip a fair coin, there are two possible outcomes: heads or tails. Each has a probability of 0.5, and thus, the sum of probabilities (0.5 for heads + 0.5 for tails) equals 1, indicating that one of these outcomes is certain to occur. In the context of histograms, this sum can be visualized as the total area of all the bars combined, ensuring that not only is each outcome's probability properly accounted for but also that the collective certainty of an outcome occurring is represented.