Problem 7 - A First Course in Probability

Question

Suppose that a die is rolled twice. What are the possible values that the following random variables can take on: (a) the maximum value to appear in the two rolls; (b) the minimum value to appear in the two rolls; (c) the sum of the two rolls; (d) the value of the first roll minus the value of the second roll?

Step-by-Step Solution

Answer

The short answer based on the given scenarios is as follows: (a) The possible maximum values of the two rolls are \( \{1, 2, 3, 4, 5, 6\} \). (b) The possible minimum values of the two rolls are \( \{1, 2, 3, 4, 5, 6\} \). (c) The possible sums of the two rolls are \( \{ 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\} \). (d) The possible values for the first roll minus the second roll are \( \{-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\} \).

Step 1: (a) Maximum value

To determine the maximum value of the two rolls, we will consider the possible outcomes when we roll the die twice. There are 6 x 6 = 36 possible outcomes, where the maximum value can vary from 1 to 6. The possible maximum values are [1, 2, 3, 4, 5, 6], which is when we have rolls like (1,1), (1,2) or (2,1), (2, 3) or (3,2), (3,4) or (4,3), (4,5) or (5,4), (5,6) or (6,5), and (6,6), respectively.

Step 2: (b) Minimum value

Similarly, to find the minimum value of the two rolls, we will consider the possible outcomes when rolling a die twice. The possible minimum values are also [1, 2, 3, 4, 5, 6], which we can get from rolls like (1,1), (1,2) or (2,1), (1,3) or (3,1), (1,4) or (4,1), (1,5) or (5,1), and (1,6) or (6,1), respectively.

Step 3: (c) Sum of the rolls

For the sum of the two rolls, we can have values ranging from 2 to 12. The possible sums are [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], which are obtained from rolls like (1,1), (1,2) or (2,1), (1,3) or (3,1), (1,4) or (4,1), (1,5) or (5,1), (1,6) or (6,1), (2,6) or (6,2), (3,6) or (6,3), (4,6) or (6,4), (5,6) or (6,5), and (6,6), respectively.

Step 4: (d) Difference between the rolls

Lastly, for the difference between the first roll and the second roll, the values can range from -5 to 5. The possible differences are [-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5], which can be obtained from rolls like (1,6), (2,6) or (1,5), (3,6) or (2,5) or (1,4), (4,6) or (3,5) or (2,4) or (1,3), (5,6) or (4,5) or (3,4) or (2,3) or (1,2), (6,6) or (5,5) or (4,4) or (3,3) or (2,2) or (1,1), (6,5) or (5,4) or (4,3) or (3,2) or (2,1), (6,4) or (5,3) or (4,2) or (3,1), (6,3) or (5,2) or (4,1), (6,2) or (5,1), and (6,1), respectively.

Key Concepts

Random Variables

When studying probability, the concept of random variables is crucial for understanding how outcomes are quantified. A random variable is a numerical description of the outcome of a random process. In other words, it's a function that assigns a numerical value to each possible outcome of a random event.

Let's take rolling a die as an example. Each face of the die represents different numerical values from 1 to 6. When you roll the die, the result can be considered a random variable because it randomly takes one of these numerical values. In our exercise, we've seen multiple random variables:

• The maximum value rolled
• The minimum value rolled
• The sum of the two rolls
• The difference between the first and second roll

Each variable provides different insights into the behavior of the dice over multiple rolls. By analyzing these variables, you can better understand the range and distribution of possible outcomes.

Dice Roll Outcomes

When you roll a die, there are six possible outcomes, each corresponding to one of the faces of the die (numbered 1 through 6). Rolling a die twice expands the range of possibilities.

If a die is rolled twice, there are 36 possible outcomes (6 outcomes for the first roll multiplied by 6 outcomes for the second roll). Some interesting questions arise, such as:

• What is the highest number rolled (maximum value)?
• What is the lowest number rolled (minimum value)?
• What is the total sum of both rolls?
• What is the difference between the two rolls?

Each of these questions corresponds to a different random variable and helps us explore the nature of dice roll outcomes. By understanding these outcomes, you get a logical way to predict and analyze other similar probabilistic events.

Mathematical Expectation

The mathematical expectation, also known as the expected value, is a concept in probability that gives a measure of the center of a probability distribution. It essentially represents the average value you would expect over numerous trials of a random experiment.

For example, the expected value of rolling a fair six-sided die is the average of all possible outcomes: \[ E(X) = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = 3.5 \]This means that if you roll the die many times, the average result should converge to 3.5.

In context with the exercise, if we were to repeatedly roll two dice, we could calculate the expectation of their sum, maximum, or minimum, offering insight into typical outcomes of these experiments. It's an essential part of making informed predictions about probabilistic events.

Distributions in Probability

Distributions in probability represent how often different outcomes happen in a random event. Probability distributions can provide insights into the behavior and likelihood of various outcomes.

For a single die roll, the probability distribution is uniform, meaning each outcome has an equal probability of occurring (1/6 for a six-sided die). However, when considering the sum of two rolls, the distribution is no longer uniform.

• The sum of the two rolls ranges from 2 to 12.
• The probability of rolling a 7 is higher because there are more combinations to produce this sum, such as (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).
• On the other hand, a sum of 2 or 12 only occurs once: (1,1) and (6,6), respectively.

Understanding these distributions helps in predicting which outcomes are more or less likely, allowing for a deeper comprehension of random events and aiding in decision-making scenarios.