Problem 100 - College Algebra with Modeling and Visualization

Question

Probability\(\quad\) A container holds \(x\) balls numbered 1 through \(x .\) Only one ball has the winning number. (a) Find a function \(f\) that computes the probability, or likelihood, of not drawing the winning ball. (b) What is the domain of \(f ?\) (c) What happens to the probability of not drawing the winning ball as the number of balls increases? (d) Interpret the horizontal asymptote of the graph of \(f\)

Step-by-Step Solution

Answer

(a) \( f(x) = 1 - \frac{1}{x} \). (b) \( x \geq 1 \). (c) It approaches 1. (d) As \( x \to \infty \), \( f(x) \to 1 \).

Step 1: Define the Probability Function

We know there are \( x \) balls and only one winning ball. The probability of drawing the winning ball is given by \( \frac{1}{x} \). Thus, the probability of not drawing the winning ball, which we denote as \( f(x) \), is \( 1 - \frac{1}{x} \). Therefore, the function \( f \) is defined as \( f(x) = 1 - \frac{1}{x} \).

Step 2: Determine the Domain of the Function

The function \( f(x) = 1 - \frac{1}{x} \) involves division by \( x \). Since division by zero is undefined, \( x \) cannot be zero. So, the domain of \( f \) is all positive integers \( x \) such that \( x \geq 1 \), because we can't have a zero or negative number of balls.

Step 3: Analyze the Probability as Number of Balls Increases

As \( x \) increases, the term \( \frac{1}{x} \) gets smaller, which makes \( f(x) = 1 - \frac{1}{x} \) approach 1. Therefore, the probability of not drawing the winning ball increases towards 1 as the number of balls increases.

Step 4: Interpret the Horizontal Asymptote

For large \( x \), the graph of \( f(x) = 1 - \frac{1}{x} \) will approach a horizontal line at \( y = 1 \). This horizontal asymptote at \( y = 1 \) indicates that as the number of balls becomes very large, the probability of not drawing the winning ball becomes very close to 1.

Key Concepts

Probability Function

When dealing with probability, the concept of a probability function is key. It helps us understand how likely an event is to happen. In our exercise, we have a container with balls numbered from 1 to \( x \), and only one ball is marked with a winning number. The task is to find out the probability of not drawing this winning ball. To calculate this, we first determine the probability of drawing the winning ball, which is \( \frac{1}{x} \). Since there is only one winning ball out of \( x \) balls, this makes sense. From here, the probability of _not_ drawing the winning ball is simply the complement: \[ f(x) = 1 - \frac{1}{x} \] This probability function, \( f(x) \), tells us how likely it is to pick any ball but the winning one.

Domain of a Function

The domain of a function refers to all the possible input values that the function can accept. In our probability function \( f(x) = 1 - \frac{1}{x} \), determining the domain is essential to ensure the function runs smoothly without errors, such as division by zero. Since we calculate \( 1 - \frac{1}{x} \), \( x \) must be greater than 0 to avoid division by zero, which is undefined in mathematics. In this context, we are dealing with a real-world scenario where ball numbers cannot be negative or zero. Thus, the smallest number of balls is 1. Therefore, the domain of \( f(x) \) is all positive integers starting from 1: - \( x \geq 1 \)- Each \( x \) represents a possible number of balls in the container.

Horizontal Asymptote

A horizontal asymptote represents a value that a function approaches but never actually reaches, as \( x \) becomes larger or smaller. In the case of the function \( f(x) = 1 - \frac{1}{x} \), as the number of balls increases, the fraction \( \frac{1}{x} \) becomes smaller and smaller. For very large values of \( x \), \( \frac{1}{x} \) approaches 0. Hence, the function value \( f(x) = 1 - \frac{1}{x} \) approaches 1. - **Horizontal Asymptote:** \( y = 1 \)This asymptote tells us that, as we have more and more balls, the probability of _not_ picking the winning ball gets closer and closer to 1, though it never quite becomes perfectly certain.

Increasing Number of Trials

In probability theory, considering an increasing number of trials helps us understand how probabilities change as conditions vary. In our scenario, increasing the number of balls (or trials) directly influences the probability function. As the number of balls, \( x \), becomes larger, the term \( \frac{1}{x} \) shrinks. Thus, the probability \( f(x) = 1 - \frac{1}{x} \) - Moves closer to 1 with each additional ball.- This implies a higher likelihood of not drawing the winning ball.This concept shows how probabilities can evolve with different scenario sizes. The more the trials, the higher the confidence in expected outcomes due to the law of large numbers helping to smooth out random fluctuations.