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Chapter 7: Problem 42

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.

Short Answer

Expert verified
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 by step solution

01

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.
02

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}.
03

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.
04

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.
05

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.

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Most popular questions from this chapter

Let \(E\) and \(F\) be events such that \(F \subset E\). Find \(P(E \mid F)\) and interpret your result.

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