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Chapter 5: Problem 3

A Monopoly player claims that the probability of getting a 4 when rolling a six-sided die is \(1 / 6\) because the die is equally likely to land on any of the six sides. Is this an example of an empirical probability or a theoretical probability? Explain.

Short Answer

Expert verified
This is an example of a theoretical probability because the player's conclusion is based not on repeated experiments or collected data, but on the logical assumption that each face of the die is equally likely to land up in a single throw.

Step by step solution

01

Understand Theoretical Probability

Theoretical Probability is the type of probability that we use when each outcome in a sample space is equally likely. It is derived purely from reasoning and logical analysis. Given a single roll of a six-sided die and asked to determine the probability of rolling a 4, each of the six outcomes (1, 2, 3, 4, 5, and 6) is equally likely. So, the probability is calculated as the number of ways that 4 can occur divided by the total number of outcomes. There is only one way to roll a 4, and there are six possible outcomes, so the theoretical probability of rolling a 4 is \(1/6\).
02

Understand Empirical Probability

Empirical probability is based on actual experiments and adequate recordings of the happening of events. This type of probability is based on the direct observations or experiences. The number of times an event happened is divided by the number of times the process was repeated. In the given scenario, the monopoly player didn't throw the dice many times and based his claim on the outcomes, hence it is not empirical probability.
03

Apply the Definitions to the Scenario

Now, applying these principles to the given scenario, it's clear that the Monopoly player鈥檚 reasoning refers to theoretical probability. The player isn't basing his statement on several dice throws, but on a logical analysis of what should happen when a six-sided die is thrown. Each of the six outcomes is equally likely in theory, regardless of what might happen in a particular run of throws. Hence, this is a theoretical probability.

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Most popular questions from this chapter

Imagine rolling a fair six-sided die three times. a. What is the theoretical probability that all three rolls of the die show a 1 on top? b. What is the theoretical probability that the first roll of the die shows a 6 AND the next two rolls both show a 1 on the top.

A jury is supposed to represent the population. We wish to perform a simulation to determine an empirical probability that a jury of 12 people has 5 or fewer women. Assume that about \(50 \%\) of the population is female, so the probability that a person who is chosen for the jury is a woman is \(50 \%\). Using a random number table, we decide that each digit will represent a juror. The digits 0 through 5 , we decide, will represent a female chosen, and 6 through 9 will represent a male. Why this is a bad choice for this simulation?

According to a recent Gallup poll, \(62 \%\) of Americans took a vacation away from home in 2017 . Suppose two Americans are randomly selected. a. What is the probability that both took a vacation away from home in \(2017 ?\) b. What is the probability that neither took a vacation away from home in \(2017 ?\) c. What is the probability that at least one of them took a vacation away from home in \(2017 ?\)

What's the probability of getting at least one six when you roll two dice? The table below shows the outcome of five trials in which two dice were rolled. a. List the trials that had at least one 6 . b. Based on these data, what's the empirical probability of rolling at least one 6 with two dice? $$ \begin{array}{|c|c|} \hline \text { Trial } & \text { Outcome } \\ \hline 1 & 2,5 \\ \hline 2 & 3,6 \\ \hline 3 & 6,1 \\ \hline 4 & 4,6 \\ \hline 5 & 4,3 \\ \hline \end{array} $$

a. How many outcomes are in the sample space? b. Assuming all of the outcomes in the sample space are equally likely, find each of the probabilities: i. all tails in 4 tosses ii. only 1 tail in 4 tosses iii. at most 1 tail in 4 tossesThe sample space given here shows all possible sequences for tossing a fair coin 4 times. The sequences have been organized by the number of tails in the sequence.
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