Question

Think about the Tiger Woods picture search again. You are opening boxes of cereal one at a time looking for his picture, which is in \(20 \%\) of the boxes. You want to know how many boxes you might have to open in order to find Tiger. a) Describe how you would simulate the search for Tiger using random numbers. b) Run at least 30 trials. c) Based on your simulation, estimate the probabilities that you might find your first picture of Tiger in the first box, the second, etc. d) Calculate the actual probability model. e) Compare the distribution of outcomes in your simulation to the probability model.

Short answer

Simulate opening boxes by using random numbers to represent finding Tiger's picture, record outcomes over 30 trials, calculate frequencies, then compare with theoretical geometric probabilities.

Step-by-step solution

Understand the Context

We are looking for Tiger Woods' picture, which is in 20% of cereal boxes. This is a problem involving success in repeated trials, similar to a geometric distribution.

Set Up the Simulation

In a simulation, we can represent each cereal box as a random number. Assign the numbers 1 to 10 for each trial, where 1 or 2 represents finding Tiger Woods' picture, and 3 to 10 represents not finding it (since 20% of boxes have the picture).

Conduct Trial Runs

Run at least 30 trials. For each trial, generate random numbers until you get a '1' or '2', which represents finding Tiger Woods' picture. Record the number of boxes opened for each trial.

Record and Analyze Simulation Data

Record the number of boxes opened in each trial. Count the frequency of finding his picture in the first box, second box, third box, etc.

Calculate Probabilities from the Simulation

Simulate the probability of finding Tiger's picture in the first box, second box, etc., based on the frequency of occurrences in your recorded data. Divide the number of successes for each position by 30 (total number of trials).

Calculate Theoretical Probability Model

Calculate the probability using the geometric distribution model: The probability of finding the picture in the first box is 0.2, in the second box is (0.8)(0.2), in the third box is (0.8)^2(0.2), and so on. This follows the formula: \[ P(k) = 0.2 \times (0.8)^{k-1} \] where k is the box number.

Compare Simulation to Theoretical Model

Compare the probabilities obtained from the simulation with the theoretical probabilities. Check for close matching and note any significant differences, which could be due to the small sample size of simulations.

Key concepts

Probability Model

A probability model is a mathematical representation of a random phenomenon. In this exercise, the probability model is used to describe the situation of finding a Tiger Woods picture in a cereal box. The probability model we're focusing on is derived from the geometric distribution, which is a type of probability distribution in statistics. It arises when we are performing experiments and are interested in the number of trials needed to get the first success.

For instance, if a Tiger Woods picture appears in 20% of the boxes, the probability of seeing the picture in the first box you open is 0.2. The probability model helps us understand and calculate the likelihoods of various outcomes based on the assumption of independence between trials.

In the context of this exercise, the probability of needing to open exactly "k" boxes before finding the picture is given by the formula: \[ P(k) = 0.2 imes (0.8)^{k-1} \] where 0.2 is the probability of success (finding the picture), and 0.8 is the probability of failure (not finding the picture) for each individual attempt.

Simulation

Simulation is a process used to imitate the operation of a real-world process or system over time. In this exercise, we use a simulation to model the scenario of opening cereal boxes to find a Tiger Woods picture.

To set up the simulation, you represent cereal boxes with random numbers. For example, assign the numbers 1 to 10 to simulate a series of trials, with numbers 1 and 2 representing a successful trial of finding the picture, and numbers 3 to 10 representing unsuccessful attempts. This is because there is a 20% chance that any given box contains the picture.

To conduct a simulation, you run multiple trials, say 30 or more, to ensure reliability. In each trial, random numbers are generated until a successful number (1 or 2) appears, which symbolizes finding the picture. The number of attempts taken in each trial is recorded, helping you observe how many boxes generally need to be opened before a picture is found.

Simulation helps visualize and understand complex probability situations, providing an empirical way to assess probabilities that might be difficult to calculate analytically.

Random Numbers

Random numbers are essential in simulations and are used to represent outcomes in random processes. In the picture-finding scenario, random numbers are assigned to boxes to determine the presence of the Tiger Woods picture. Each random number simulates the result of opening a box.

In our case, the numbers 1 and 2 are chosen to signify success—finding the picture—and numbers 3 through 10 indicate failure—not finding the picture. This gives us the flexibility to model the 20% probability of success per box.

• Numbers 1 and 2 represent that the box contains the picture (20% probability).
• Numbers 3 to 10 indicate that the box does not contain the picture (80% probability).

Using random numbers allows us to repeatedly simulate the process, ensuring that we capture the randomness inherent in the real scenario. Computers can easily generate these numbers using algorithms, ensuring the randomness needed for reliable simulation outcomes.

Understanding how to use random numbers is crucial as it lays the foundation for running accurate and efficient simulations.

Probability Theory

Probability theory is the branch of mathematics that deals with the likelihood of different outcomes. It is the foundation upon which both the probability model and simulations are built. It explains how and why certain results occur and what to expect when performing experiments involving chance.

In this exercise, we utilize probability theory to understand both the empirically obtained probabilities from our simulations and the theoretically calculated probabilities from our geometric distribution model. This dual approach allows us to compare and contrast real-world experimental data with mathematical expectations.

The probability theory helps us calculate the likelihood of events accurately and provides tools to predict outcomes with precision. By using geometric distribution in this scenario, probability theory also enables us to calculate the chances of needing "k" boxes to find the first Tiger Woods picture. These calculated probabilities can be plotted to create a probability distribution, offering a visual insight into the problem at hand.

In summary, probability theory allows us to move beyond intuition and provide numeric estimates of how many boxes might be needed to find a picture, offering both theoretical insights and practical applications.