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Chapter 5: Problem 49

Use a graphing calculator or statistical software to simulate rolling a six- sided die 100 times, using an integer distribution with numbers one through six. (a) Use the results of the simulation to compute the probability of rolling a one. (b) Repeat the simulation. Compute the probability of rolling a one. (c) Simulate rolling a six-sided die 500 times. Compute the probability of rolling a one. (d) Which simulation resulted in the closest estimate to the probability that would be obtained using the classical method?

Short Answer

Expert verified
The closest probability estimate to the theoretical value of \(\frac{1}{6}\) was found in the simulation with 500 rolls.

Step by step solution

01

- Set Up the Simulation for Rolling the Die 100 Times

Use a graphing calculator or statistical software to create a simulation that rolls a six-sided die 100 times. Make sure it generates random integers between 1 and 6.
02

- Record and Analyze the Results

Run the simulation and record the number of times each number (1 through 6) appears. For part (a), calculate the frequency of rolling a one and divide it by 100 to find the probability.
03

- Repeat the Simulation

Repeat the same simulation (rolling the die 100 times) to verify the consistency of your result. Compute the frequency of rolling a one again and divide by 100.
04

- Simulate Rolling the Die 500 Times

Change the settings in your calculator or software to simulate rolling a six-sided die 500 times. Record the results for each number (1 through 6), paying attention to the frequency of rolling a one.
05

- Calculate the New Probability

Compute the frequency of rolling a one in the 500-roll simulation results and divide it by 500 to find the new probability.
06

- Compare Simulations

Evaluate which simulation鈥檚 probability is closest to the theoretical probability of rolling a one, which is \(\frac{1}{6}\) or approximately 0.1667.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

graphing calculator
A graphing calculator is a powerful tool that can perform many types of calculations, including simulations. For this exercise, you'll use it to simulate rolling a six-sided die. The calculator can generate random integers within a specified range, making it perfect for this task. Throughout the simulation, the results can be easily recorded and analyzed directly on the device.
integer distribution
An integer distribution refers to a probability distribution where the outcomes are integers. In this exercise, each roll of a six-sided die results in an integer between 1 and 6. The idea is to generate a sufficient number of these random integers to simulate real-world events. As you roll the die multiple times, you can observe how often each number appears, giving you a distribution of the possible outcomes.
frequency analysis
Frequency analysis involves counting the number of times each outcome occurs in your simulation. For example, you might roll the die 100 times and get the following results: 15 times a '1', 18 times a '2', and so on. By analyzing these frequencies, you can estimate the probabilities of each outcome. To find the probability of rolling a '1', divide the number of times '1' appears by the total number of rolls.
theoretical probability
The theoretical probability of an event is based on the expected outcomes in a perfectly controlled environment. For a six-sided die, the theoretical probability of rolling any specific number is \(\frac{1}{6}\) or approximately 0.1667. By comparing your simulation results to this theoretical value, you can assess the accuracy of your simulation. Multiple trials in the simulation tend to produce results that converge towards this theoretical probability.
random integers
Random integers are numbers that occur in a sequence, where each number has an equal chance of occurring. In the context of rolling a die, each integer between 1 and 6 should appear with equal probability. This randomness is crucial for simulating realistic results. By using a graphing calculator or statistical software to generate these random integers, you can accurately mimic the process of rolling a die, ensuring a fair simulation.

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Most popular questions from this chapter

The following data represent, in thousands, the type of health insurance coverage of people by age in the year 2002 $$\begin{array}{llllll}\hline & <18 & 18-44 & 45-64 & >64 \\\\\hline \text { Private } & 49,473 & 76,294 & 52,520 & 20,685 \\\\\hline \text { Government } & 19,662 & 11,922 & 9,227 & 32,813 \\\\\hline \text { None } & 8,531 & 25,678 & 9,106 & 258 \\\\\hline\end{array}$$ (a) What is the probability that a randomly selected individual who is less than 18 years old has no health insurance? (b) What is the probability that a randomly selected individual who has no health insurance is less than 18 years old?

Suppose a single card is selected from a standard 52-card deck. What is the probability that the card drawn is a king? Now suppose a single card is drawn from a standard 52-card deck, but we are told that the card is a heart. What is the probability that the card drawn is a king? Did the knowledge that the card is a heart change the probability that the card was a king? What is the term used to describe this result?

List all the combinations of four objects \(a, b, c,\) and \(d\) taken two at a time. What is \(_{4} C_{2} ?\)

Suppose a compact disk (CD) you just purchased has 13 tracks. After listening to the CD, you decide that you like 5 of the songs. With the random feature on your CD player, each of the 13 songs is played once in random order. Find the probability that among the first two songs played (a) You like both of them. Would this be unusual? (b) You like neither of them. (c) You like exactly one of them. (d) Redo (a)-(c) if a song can be replayed before all 13 songs are played (if, for example, track 2 can play twice in a row).

The following data represent the number of driver fatalities in the United States in 2002 by age for male and female drivers: $$\begin{array}{|l|c|c|} \hline \text { Age} & \text { Male } & \text { Female }\\\\\hline \text { Under } 16 & 228 & 108 \\\\\hline 16-20 & 5696 & 2386 \\\\\hline 21-34 & 13,553 & 4148 \\\\\hline 35-54 & 14,395 & 5017 \\\\\hline 55-69 & 4937 & 1708 \\\\\hline 70 \text { and over } & 3159 & 1529 \\\\\hline\end{array}$$ (a) What is the probability that a randomly selected driver fatality who was male was 16 to 20 years old? (b) What is the probability that a randomly selected driver fatality who was 16 to 20 was male? (c) Suppose you are a police officer called to the scene of a traffic accident with a fatality. The dispatcher states that the victim is 16 to 20 years old, but the gender is not known. Is the victim more likely to be male or female? Why?
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