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Designing Simulations

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Explore the Monty Hall Problem through a hands-on team activity where you will design simulations to test probabilistic outcomes. Break into pairs, complete the tasks on page 464, fill in the chart for number 5, and answer questions 6 and 7. Understand key concepts such as simulations, trials, and Monte Carlo methods. Engage in practical examples, including throwing darts randomly and using random numbers to simulate trials. Collect data, estimate probabilities, and deepen your understanding of this classic probability puzzle. ###

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Designing Simulations

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  1. Designing Simulations Section 7.7

  2. Monty Hall Problem • Break up into teams of 2 and complete the activity on page 464. Fill in the chart for number 5, and answer questions 6 and 7.

  3. Definitions • Simulation: a way to test the results of an experiment without actually conducting the experiment. • Trial: each repetition of the experiment

  4. Monte Carlo Simulations • 1. determine how it will be simulated • 2. Define what constitutes a single trial and what data will be recorded. • 3. Specify the number of trials that will be run and how the estimated answer will be obtained. **very helpful – p. 874 (random numbers) BEGIN AT ANY NUMBER(not beginning)!!!

  5. Example 1 • Four darts are thrown randomly at a coordinate plane and the quadrants are recorded. It is desired to simulate the probability that all 4 darts land in at most 2 quadrants. Design a simulation with 50 trials to estimate this probability. Sample: roll two dice, 1 – 4 stand for quadrants. Re-roll if it lands on 5 or 6. Roll die 4 times – record answer. Repeat 50 times.

  6. Example 2 • Explain how to use a table of random numbers to simulate the trials in Example 1. Take the numbers in pairs. If the number is between 0 – 24  quad 1 25 – 49  quad 2 50 – 74  quad 3 75 – 99  quad 4 Look at 8 numbers. Record results. Repeat 50 times!

  7. Example 3 • Run the simulation mentioned in example 2. I began my random number at row 23, column 10. 1, 4, 1, 3 1, 3, 1, 1 4, 4, 1, 4 … 18 / 50 = .36 Theoretical probability: 1/16 + ¼ + 1/16 = 3/8

  8. Homework Page 469 3, 4, 6 – 8 (skip 8 b)

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