Question

Driving test. You are about to take the road test for your driver's license. You hear that only \(34 \%\) of candidates pass the test the first time, but the percentage rises to \(72 \%\) on subsequent retests. Estimate the average number of tests drivers take in order to get a license. Your simulation should use at least 20 runs.

Short answer

Approximately 2-3 tests are needed on average.

Step-by-step solution

Understand the Problem

We need to simulate the process of taking driving tests multiple times until passing. Initially, there is a 34% pass rate, and for each subsequent attempt, the pass rate increases to 72%. Our goal is to find the average number of tests taken over at least 20 simulations.

Define Simulation Parameters

We have two probabilities: P(pass on first attempt) = 0.34, and P(pass on subsequent attempts) = 0.72. We will perform 20 simulations to ensure statistical validity.

Execute the Simulation

For each simulation, generate a random number between 0 and 1 to represent the outcome of the test attempt. If the number is less than or equal to the pass probability, consider the test passed. Record how many attempts are needed for each simulation to pass the test.

Determine the Passing Condition

In each run, for the first attempt, use a probability of 0.34. If the test is not passed on the first try, continue testing using a probability of 0.72 for all subsequent attempts until a pass is achieved.

Calculate the Average Number

Sum the total number of attempts from all 20 simulations, then divide by 20 to find the average number of tests taken. This provides an estimate of the average number of tests required to pass.

Example Calculation

Suppose one simulation run results in the attempts: pass on first, fail then pass, pass on third, fail fail then pass, etc. Calculate the total number of attempts like: 1 (for first pass) + 2 (one fail, one pass) + 1 (pass) + 3 (two fails, one pass) = 7 for four example runs.

Evaluate Results

After running the simulation 20 times, compile the results to find the average. Ensure that each run's number of attempts is included. Consider variance to understand fluctuations in sample runs.

Key concepts

Probability

Probability is a fundamental concept in statistics that measures how likely an event is to occur. In the context of our driving test problem, probability helps us predict the chances of passing the test on various attempts.

To start, when you take the driving test the first time, there is a 34% chance of passing. This probability is represented as a decimal, which is 0.34. It is crucial to understand that this number illustrates a part of a whole, or how many times out of 100 we expect the event (passing the test) to happen.

Let's break down probabilities:

• A probability of 0 or 0% means the event will not happen.
• A probability of 1 or 100% means the event will always happen.
• Probabilities between 0 and 1 show varying likelihoods.

After the first attempt, the probability of passing increases to 72% or 0.72. This change reflects a greater chance of success on following attempts, which might be due to experience gained or the reduction of nervousness. By understanding these probabilities, we can better anticipate the potential outcomes of test attempts.

Statistical Validity

Statistical validity is about ensuring that the results of your simulation or experiment reflect true patterns or facts of the real world. When we run multiple simulations, we want our findings to be reliable, credible, and applicable in real-life situations.

In the driving test simulation, we strive for statistical validity by performing at least 20 runs. Why 20?

• A higher number of runs leads to more data, which typically results in more stable and accurate averages.
• Variance, which is the measure of how data points differ from the average value, is minimized with more trials.
• This sample size helps smooth out anomalies or unusual results that might occur if we only did, say, three or four tests.

By applying this number of simulations, we can ensure our average reflects a meaningful trend rather than random chance.

Average Calculation

Average calculation is an essential method in statistics that helps summarize large amounts of data into a single, representative figure. In our driving test simulation, the average number of tests taken plays a critical role.

To calculate the average, we follow a multi-step process:

• Record Each Simulation: Keep track of how many tests it takes in each simulation trial until a pass occurs.
• Sum the Total Tests: Add all the attempts required from each of the 20 simulation runs.
• Divide by Number of Simulations: Finally, divide the total number of test attempts by 20 (the number of simulations) to find the average.
  

The average provides an estimate of how many attempts, on the whole, a candidate might expect to take to pass the test. It simplifies complex results into a single most-typical value, providing a clear, concise answer to a potentially confusing question.