Problem 1 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Three balls are selected at random without replacement from an urn containing four green balls and six red balls. Let the random variable \(X\) denote the number of green balls drawn. a. List the outcomes of the experiment. b. Find the value assigned to each outcome of the experiment by the random variable \(X\). c. Find the event consisting of the outcomes to which a value of 3 has been assigned by \(X\).

Step-by-Step Solution

Answer

a. The possible outcomes are: GGG, GGR, GRG, GRR, RGG, RGR, RRG, RRR. b. The values assigned by the random variable \(X\): - GGG: \(X = 3\) - GGR, GRG, RGG: \(X = 2\) - GRR, RGR, RRG: \(X = 1\) - RRR: \(X = 0\) c. The event with \(X = 3\) is {GGG}.

Step 1: List all possible outcomes

Since we are drawing three balls, each draw can result in either a green ball (G) or a red ball (R). We can list the possible outcomes as follows: 1. GGG 2. GGR 3. GRG 4. GRR 5. RGG 6. RGR 7. RRG 8. RRR

Step 2: Assign values to outcomes by the random variable \(X\)

The random variable \(X\) represents the number of green balls drawn. We can assign values to the outcomes listed in Step 1 based on this: 1. GGG: \(X = 3\) 2. GGR: \(X = 2\) 3. GRG: \(X = 2\) 4. GRR: \(X = 1\) 5. RGG: \(X = 2\) 6. RGR: \(X = 1\) 7. RRG: \(X = 1\) 8. RRR: \(X = 0\)

Step 3: Find the event with \(X = 3\)

Now, we need to find the event consisting of the outcomes to which a value of 3 has been assigned by \(X\). From the values assigned in Step 2, we can see that there is only one outcome with \(X = 3\): 1. GGG: \(X = 3\) Thus, the event to which a value of 3 has been assigned by \(X\) is {GGG}.

Key Concepts

Random Variables

A random variable is a fundamental concept in probability and statistics. In simple terms, it's a way to quantify the outcomes of a random experiment. A random variable assigns a numerical value to each possible outcome of the experiment.
In our exercise, the random variable is represented by the letter \( X \). Specifically, \( X \) counts the number of green balls drawn from the urn. When we talk about the value assigned to an outcome by a random variable, we're referring to how the random experiment's result translates into a number.

• For "GGG" (all green balls), \( X = 3 \) because three green balls are drawn.
• For "GGR" (two green, one red), \( X = 2 \).
• Continuing this pattern, for "RRR" (no green balls), \( X = 0 \).

The role of random variables is to help us understand and work with probabilities and distributions by simplifying and categorizing the outcomes into meaningful numbers.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations of objects. It helps us count how many different ways we can pick, arrange, or combine objects.
In problems like selecting balls from an urn, combinatorics can help determine the number of possible outcomes. When the exercise states that three balls are drawn from an urn, combinatorial techniques help us list all potential outcomes.
For example, if there are four green balls (G) and six red balls (R), drawing three can result in combinations like "GGG," "GGR," "GRG," and so on. Combinatorics provides the methods to ensure every combination is considered, such as if the order of selection matters or doesn't. In this problem, the order matters since we're not replacing each ball after it's drawn.
The application of combinatorics ensures that probabilities are calculated accurately by accounting for every possible sequence without overlooking any potential combination of outcomes.

Discrete Probability

Discrete probability deals with the likelihood of each outcome in a finite set of possibilities. Our exercise is an example of discrete probability because there is a finite number of outcomes when drawing balls from an urn.
Discrete probability tells us how often each outcome is expected to occur. Given all possible outcomes, we assign probabilities based on each outcome’s likelihood. In this exercise:

• The probability of drawing "GGG" depends on the initial number of green balls and the calculation of drawing without replacement.
• Similarly, the probability of outcomes like "GGR" or "RGR" reflects the changing probability as balls are drawn and removed from the urn.

Each outcome in the list of possible draws has an associated probability, and understanding discrete probability helps us make predictions or decisions based on these calculations. This concept forms the basis for more complex probability theories and applications in various fields.