Problem 2 - Calculus for Biology and Medicine

Question

In Problems \(1-4\), determine the sample space for each random experiment. The random experiment consisting of rolling a six-sided die twice.

Step-by-Step Solution

Answer

The sample space consists of 36 pairs: \((1,1), (1,2), \ldots, (6,6)\).

Step 1: Understand the Random Experiment

In this problem, the random experiment consists of rolling a six-sided die twice. Each roll is independent and can result in one of six outcomes: 1, 2, 3, 4, 5, or 6.

Step 2: Define the Sample Space

The sample space is the set of all possible outcomes. Since a six-sided die is rolled twice, the possible outcomes for each roll are combined. Thus, the sample space consists of pairs \((x, y)\), where \(x\) is the result of the first roll and \(y\) is the result of the second roll. Both \(x\) and \(y\) can be any number from 1 to 6.

Step 3: Calculate the Total Number of Outcomes

Each roll of the die has 6 possible outcomes. Since the rolls are independent, the total number of outcomes is the product of the possibilities from each roll: 6 (for the first roll) times 6 (for the second roll), resulting in \(6 \times 6 = 36\) possible outcomes.

Step 4: List the Sample Space

The sample space can be represented as all possible ordered pairs: \((1,1), (1,2), (1,3), (1,4), (1,5), (1,6), \ldots, (6,6)\). This includes every combination of the results from the first and second die rolls, resulting in a total of 36 pairs.

Key Concepts

Understanding Random Experiments

A random experiment is a process that leads to one of several possible outcomes. In probability, these experiments are not predictable in advance but have a well-defined set of possible outcomes. Consider rolling a six-sided die. The die can land showing any integer between 1 to 6. When rolling it twice, each roll constitutes a separate event. Thus, the random experiment here involves these two rolls, each yielding one of six possible results. Hence, each round of rolling offers an entirely new possibility.

Defining Independent Events

Independent events occur when the outcome of one event does not affect the outcome of another. In probability, events are considered independent if the occurrence of one event has no impact on the likelihood of another event occurring.

For example, when rolling a six-sided die, the outcome of the first roll doesn't affect the outcome of the second roll. You have a 1/6 chance to roll any given number on each throw. Multiplying these probabilities confirms the outcomes remain distinct and unaffected by each other. These are core principles in probability, used to define sample spaces and calculate overall probabilities.

Rolling a Die and Its Sample Space

Rolling a die is a common example in probability discussions, thanks to its simplicity and limited number of outcomes. A standard die has six numbers, from 1 to 6.

When you roll a die twice, forming a complete set of outcomes means you pair each result from the first roll with each result from the second roll. This creates ordered pairs like \( (1,1) \) and \( (6,6) \), forming a complete sample space. Since a die roll represents independent events, you multiply the number of possible outcomes from each roll. So, with 6 possible results per roll, you get \( 6 \times 6 = 36 \) outcomes total. This example illustrates how to build a sample space for similar experiments, using foundational probability concepts.