## Computer Simulation

To get a better understanding of the random process involved here, we should look at many more trials of the process, under identical conditions. We will turn to technology to do so. The applet to the right (or click here) will allow you to select a strategy and play the game thousands of times.

(e) Suppose that you were to play this game 1,000 times. In what proportion of those games would you expect to win the car? Explain.

(f) Use the applet to simulate playing this game 10 times with the Stay strategy:

• Click on a door and then press that door again to stay.
• The computer will reveal whether you won or lost and will keep track
• Press on any of the doors again or press the Play Again button and repeat until you have played ten times

Record the percentage of wins in these 10 plays with the Stay strategy in the table in your lab report.

• Now have the computer play 10 more times with the Stay strategy by pressing the Go button. Record the percentage of wins in these 20 games in the table in your lab report.
• Then press Go again to simulate another 10 games, and record the overall percentage of wins at this point (after 30 games). Keep doing this in multiples of 10 games until you reach 100 total games played with the Stay strategy.
Record the overall percentages of wins after each additional multiple of 10 games in the table in your lab report.

 Number of games 10 20 30 40 50 60 70 80 90 100 Percentage of wins

(g) Now uncheck the Animate simulation box and change the times to play from 10 to 1000 and press Go. Report the resulting percentage of wins.

 Number of games 100 200 300 400 500 600 700 800 900 1,000 Percentage of wins

(h) What do you notice about how the percentage of wins changes as you play more games? Does this percentage appear to be approaching some common value?

 Definition: The probability of an outcome refers to the long-run proportion of times that the outcome would occur if the random process were repeated a very large number of times under identical conditions. You can approximate a probability by simulating the process many times. Simulation leads to an empirical estimate of the probability, which is the proportion of times that the event occurs in the simulated repetitions of the random process.

(i) Based on your simulation results so far, what do you estimate for the probability of winning the Monty Hall game if you use the Stay strategy? [Hint: Keep in mind that probabilities should always be expressed as decimal values between 0 and 1.]

(j) How could you use the simulation to obtain a better estimate of this probability?