Question

You are one space short of winning a child's board game and must roll a 1 on a die to claim victory. You want to know how many rolls it might take. a) Describe how you would simulate rolling the die until you get a 1 . b) Run at least 30 trials. c) Based on your simulation, estimate the probabilities that you might win on the first roll, the second, the third, etc. d) Calculate the actual probability model. e) Compare the distribution of outcomes in your simulation to the probability model.

Short answer

Run a simulation of die rolls 30 times, calculate the frequency of rolling a 1 for each scenario, and compare these results to theoretical probabilities.

Step-by-step solution

Simulating the Die Roll

To simulate rolling a die until you get a 1, use a random generator function that produces integers between 1 and 6. Repeat the roll until the result is 1 and count how many rolls it takes for each trial.

Running 30 Trials

Perform the simulation in Step 1 a total of 30 times. Record the number of rolls it takes to get a 1 for each of these 30 trials. This data will be used for probability estimates.

Estimating Probabilities

Calculate the frequency of getting a 1 on the first roll, second roll, etc., from your 30-trial dataset. Divide the number of successes by the total number of trials to estimate the probabilities for each scenario (winning on first roll, second roll, etc.).

Calculating the Probability Model

For a fair six-sided die: the probability of rolling any specific number (like a 1) is \( \frac{1}{6} \). The probability of winning on the first roll is \( \frac{1}{6} \), on the second roll is \( \left( \frac{5}{6} \right) \times \left( \frac{1}{6} \right) \), because you must not roll a 1 on the first roll and then a 1 on the second roll, and so on.

Comparing Simulation and Probability Model

Compare your simulation probabilities from Step 3 with the calculated probabilities from Step 4. Note any discrepancies and consider factors like variance due to limited trials or imperfect randomness in your simulation.

Key concepts

Random Generator Function

To simulate the roll of a die, we need a random generator function. This function generates random numbers within a specified range. For a standard six-sided die, the range is 1 to 6.

In practice, this function mimics the randomness of throwing a physical die. Each roll is independent, and each number should have an equal chance of appearing.

In computational terms, we can use functions found in many programming languages, like `randint(1, 6)` in Python. This command will simulate one roll of a six-sided die.

To successfully complete the simulation, you must repeatedly use this function until you obtain a 1. Count how many rolls it takes in each trial to achieve this result. This setup is essential for gathering data for further analysis.

Estimating Probabilities

Once the data from multiple trials is ready, it's time to estimate probabilities. Start by noting how often each outcome occurs across the trials.

• For instance, track how many times you win on the first roll, second roll, and so on.
• Divide the number of times each outcome occurs by the total number of trials to estimate its probability.

This process helps transform raw data into meaningful probability estimates, from which you can interpret the likelihood of each event. If, in your dataset of 30 trials, you win on the first roll 5 times, the estimated probability for winning on the first roll is calculated as: \[ P(\text{win on first roll}) = \frac{5}{30} = \frac{1}{6} \]

These estimates provide insight into how likely each scenario is based on your simulation.

Probability Model

The next step is calculating the probability model. For our dice scenario, the probability of rolling a specific number on a fair six-sided die is always \( \frac{1}{6} \). This remains constant for each roll because each side is equally likely.

To determine the probability of winning on subsequent rolls, you must consider the sequences of previous non-winning rolls. For example:

• To win on the second roll: first roll not a 1, second roll a 1.
• Probability: \( \left( \frac{5}{6} \right) \times \left( \frac{1}{6} \right) \).

This rule extends with similar logic for further rolls. Thus, the probability model gives a mathematical framework to predict outcomes based on ideal conditions.

Distribution of Outcomes

Finally, let's compare the distribution of outcomes from the simulation with the theoretical probability model.

The distribution of outcomes in your simulation is all about frequency and how many times each win condition occurs across your trials. It may vary slightly from our theoretical predictions due to randomness and limited sample size.

Use a visual or numerical comparison to analyze:

• Check if results like winning on the first or second roll align with expected probability model probabilities.
• Consider factors like sample variance or anomalies.
• Understand that real-life randomness might slightly differ from ideal mathematical models.

By comparing these, you'll see how well your simulation approximates true probability models, offering valuable insights into both practical and theoretical aspects of probability.