Problem 42 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the given experiment has a sample space with equally likely outcomes. A loaded die is rolled, and the number appearing uppermost on the die is recorded.

Step-by-Step Solution

Answer

The experiment of rolling a loaded die and recording the number appearing uppermost does not have a sample space with equally likely outcomes. This is because the die is loaded, resulting in an altered probability distribution with some outcomes being more likely than others.

Step 1: Understanding a Loaded Die

A loaded die is a die that has been tampered with to make certain outcomes more likely than others. This means that the probabilities for each face are not equal, as in a fair die. The probability distribution for a loaded die will have different values for each outcome.

Step 2: Identifying the Sample Space

The sample space for this experiment is the set of all possible outcomes. When rolling a six-sided die, there are six possible outcomes: {1, 2, 3, 4, 5, 6}.

Step 3: Calculating Probabilities for Loaded Die

As we are only given information about the die being loaded, and not the specific probability distribution, we cannot determine the exact probabilities for each outcome. Instead, we will consider the general properties of a loaded die. Since the die is loaded, it has been altered in a way that makes some outcomes more likely than others. This implies that not all faces have an equal probability of appearing.

Step 4: Comparing Probabilities

Based on the information given (the die is loaded), we can conclude that at least one face has a different probability than others. In this case, the probabilities are not equal, and the sample space is not composed of equally likely outcomes.

Step 5: Conclusion

The experiment of rolling a loaded die and recording the number appearing uppermost does not have a sample space with equally likely outcomes. This is due to the fact that the die is loaded, altering the probability distribution and making some outcomes more likely than others.

Key Concepts

Loaded Die

When we talk about a loaded die, we are referring to a die that has been intentionally altered to change the probabilities of each of its possible outcomes. This alteration means that some numbers are more likely to appear than others. Unlike a fair six-sided die, where each face has an equal probability of coming up (1/6), a loaded die does not have uniform probabilities.
A loaded die can be tampered with in many ways, such as adding weight to one side or slightly modifying its shape, to favor certain numbers. The key point to remember is that these changes disrupt the equal likelihood of outcomes, making the die unpredictable and biased.

• Loaded dice can be used in games to cheat.
• They are often used as examples in probability theory to illustrate biased outcomes.

Sample Space

In probability theory, the sample space is the complete set of all possible outcomes of an experiment. For a standard six-sided die, the sample space comprises the numbers {1, 2, 3, 4, 5, 6}. Each of these numbers corresponds to a potential outcome when the die is rolled.
The sample space provides the foundation for calculating probabilities and understanding how different outcomes relate to one another. Understanding the sample space is crucial because it defines the scope of what can happen in any given experiment.

• A complete sample space lists every possible outcome.
• In the context of dice, the sample space remains the same whether the die is loaded or fair.

Equally Likely Outcomes

Equally likely outcomes are a fundamental concept in probability theory. This occurs when every possible outcome of an experiment has the same chance of happening. In the context of a fair six-sided die, each number (1 through 6) is equally likely to appear, with a probability of 1/6.
However, when dealing with loaded dice, the outcomes are not equally likely. The modification of the die affects the probabilities, making some outcomes more probable than others. Understanding whether outcomes are equally likely helps determine whether the die or any experiment is fair or biased. This is critical in both game settings and statistical analysis.

• In a fair die: all outcomes have equal probability.
• In a loaded die: the probability of outcomes is altered, disrupting fairness.