You are about to take the road test for your driver’s...

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. a) Create a plan for a simulation to estimate the average number of tests drivers take in order to get a license. b) Here are results of 100 trials of a simulation. Use these results to estimate the average number of tests drivers take in order to get a license.

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.
a) Create a plan for a simulation to estimate the average number of tests drivers take in order to get a license.
b) Here are results of 100 trials of a simulation. Use these results to estimate the average number of tests drivers take in order to get a license.

Key Concepts

Geometric Distribution

The geometric distribution models the number of independent trials needed to achieve the first success in a sequence of Bernoulli trials. Although the scenario may involve varying probabilities, the concept helps in understanding the structure of the simulation related to success after multiple attempts.

Discrete Probability Distributions

Discrete probability distributions model scenarios where outcomes are countable, such as the number of tests taken until passing. They are fundamental in setting up the simulation and calculating the likelihood of each possible number of tests.

Simulation Experiment Design

Designing a simulation experiment involves planning the procedure to mimic a real-life process using random sampling. This includes defining the test process, setting initial and subsequent probabilities, running many independent trials, and aggregating the outcomes to estimate metrics like the average number of tests required.

Monte Carlo Simulation

A Monte Carlo simulation is a computational technique that uses repeated random sampling to model complex systems and processes. In this context, it allows for estimating the average number of tests taken by simulating the random outcomes of each test and averaging the results over many trials.

Expected Value

The expected value is a key concept in probability, representing the long-run average outcome of a random variable. In simulation studies, it is used to determine the average number of tests required by weighting each possible outcome with its probability.

Transcript

00:01 So you have a driver's test and you know that when you hear that 34 % of the candidates will pass on the first time and so that means that 66 % will fail and then these that pass won't take another test but these have a higher rate of now passing the second time.

00:19 If there's a 72 % chance of passing and then there is a 28 % chance of failing.

00:26 Now you want to create a plan for simulation to estimate the average number of time of drivers that it takes in order to pass the test and then again we would continue to go here if you fail you get 72 % chance of passing and then here and so this could continue on where this branch keeps going.

00:55 So what we do is we could start out with generating a random number and have rand int and generate numbers from 1 to 100.

01:11 Whoops 1 comma 100 and let's get rid of that little space and this will be the first attempt and then we will let 1 through 34 stand for you pass and then means it took one time and we'll let 35 through 100 stand for fail and then after that if it's a fail then you use rand int again of 1 to 100 however this time we're gonna let 1 through 72 stand for a pass and 35 excuse me 73 through 100 stand for a fail and then if it's a fail if it's a pass here then it takes two attempts here it takes one attempt and here it would end up being continuing on if this then you continue repeat number two until you get a p.

02:26 Now it says use the results to find the average number and to do a number of trials and you may have a slicker way of doing this but i would have to keep hitting that rand int and then keep track of trials.

02:38 So i'm going to get that feature on my calculator rand int and i'll tell you what my first result is my first result and let's change to blue my first trial one i have a 44 which means that i did not pass so now i have to hit the button again and the second number is a 17 which means it took two attempts to pass.

03:07 Second trial hit the button again we have 35.

03:12 35 is not passing so we need to have another number and we have 50 and that is a pass.

03:19 Third trial hit the number i get 96 which means didn't pass and then hit 75 and once again didn't pass...

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