Question

Suppose \(75 \%\) of all drivers always wear their seatbelts. Let's investigate how many of the drivers might be belted among five cars waiting at a traffic light. a) Describe how you would simulate the number of seatbelt-wearing drivers among the five cars. b) Run at least 30 trials. c) Based on your simulation, estimate the probabilities there are no belted drivers, exactly one, two, etc. d) Find the actual probability model. e) Compare the distribution of outcomes in your simulation to the probability model.

Short answer

Simulate with trials, calculate approximate and exact probabilities, then compare both distributions.

Step-by-step solution

Simulation Setup

To simulate the number of seatbelt-wearing drivers among five cars, we will assign each driver a probability of 0.75 of wearing a seatbelt. This can be done using a random number generator or by rolling a 4-sided die. If the random outcome is one of the first three options (75%), we count the driver as wearing a seatbelt. Otherwise, they are counted as not wearing a seatbelt.

Run 30 Trials

Conduct at least 30 repetitions of our simulation where each trial consists of assessing the seatbelt status of 5 drivers.

Tabulate Simulation Results

For each trial, record the number of seatbelt-wearing drivers (from 0 to 5). This will give us a count for each possible outcome across all trials, allowing us to estimate the probability for each count.

Calculate Simulation-Based Probabilities

Divide the number of times each outcome (0, 1, 2, 3, 4, or 5 belted drivers) occurred by the total number of trials (30) to estimate probabilities.

Determine Actual Probability Model (Binomial Distribution)

Use the binomial probability formula to calculate the exact probabilities for each potential outcome: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]where \( n = 5 \), \( p = 0.75 \), and \( k \) is the number of successes (belted drivers). Calculate this for each \( k \) from 0 to 5.

Compare Simulation to Actual Probability Model

Compare the estimated probabilities from the simulation with those calculated using the binomial distribution. Look for similarities and differences in the distribution of outcomes.

Key concepts

Binomial Distribution

When exploring probability scenarios involving two potential outcomes, such as success or failure, the binomial distribution is a useful statistical tool. In the context of the seatbelt problem, each driver is a trial with:

• Success: The driver is wearing a seatbelt.
• Failure: The driver is not wearing a seatbelt.

The number of trials is fixed, set at 5 cars, with each having a 75% chance of success. The binomial distribution can mathematically express this scenario using the formula:\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]where:

• n is the total number of trials (5 cars).
• k is the number of successful outcomes (belted drivers).
• p is the probability of success (0.75).

This formula calculates the probability of having exactly k belted drivers among 5. The binomial distribution provides exact probabilities for each possible number of successes (from 0 to 5 in this case). It lays the foundation for comparing results derived from actual probabilities versus those estimated from simulations.

Probability Model

A probability model is a mathematical description of a random phenomenon. In this scenario, we need a model to predict outcomes about drivers wearing seatbelts. A good probability model allows us to determine the likelihood of various outcomes and informs us of expected variability in our predictions.

• For our seatbelt problem, the probability model relies on the binomial distribution.
• Each driver is considered independent.
• The probability (0.75) remains consistent across trials.

With this model, we can precisely calculate the chances for 0 to 5 belted drivers amongst five cars. Moreover, by providing a structured understanding of potential outcomes and their probabilities, the model assists in validating simulation data. When we compare simulation results to the probability model, we can evaluate their accuracy and make informed predictions.

Simulation Trials

Simulation trials offer a practical means of approximating outcomes when exact calculations are impractical or cumbersome. They involve repeating an experiment (like checking seatbelt usage among drivers) multiple times.
Conducting simulation trials consists of a few critical steps:

• Setup: Define the conditions and probabilities; here, each driver has a 0.75 chance of wearing a seatbelt.
• Execution: Run multiple trials (at least 30 in this case) to generate data that reflects potential real-world outcomes.
• Data Collection: For each trial, note how many drivers are wearing seatbelts.

Through this method, the probability of having a certain number of belted drivers is estimated based on the relative frequency of observed results. Such probabilistic simulations provide a tangible way to visualize potential outcomes and gather insights that align with theoretical models, like the binomial distribution. Besides, simulations offer opportunities to explore scenarios without a need for complex calculations, enhancing comprehension and practical understanding.