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 |
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(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 |
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(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?
The probability of winning with the Stay stragtegy being 1/3 makes sense, because your chance of picking the correct door to begin with is one-out-of-three. Many people believe the probability of winning with the Switch strategy is 0.5 because there are now just two doors. Our simulation tools give us a very easy way to investigate this.
- Return to the Monty Hall applet and use the pull-down menu to now use the "Switch" strategy.
- Set the number of times to 1000
- Uncheck the Animate simulation box
- Press the Go button
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When you create output like this, we will often want you to save your results. The simplest way to make a “screen capture” of your results is to use the “Snipping Tool” (see the technology notes in PolyLearn for a couple of other options). To use the Snipping Tool, click on the scissors icon along the bottom toolbar. If it’s not there, select Start > All Programs > Accessories > Snipping Tool. The tool will open with the New button already pressed. Now drag your mouse to select the region you want to copy (the table and the Cumulative Win Proportion graph). This will display in a preview window. Use ctrl-C/ctrl-V to copy and paste this image into this Word file.
(k) Use the Snipping Tool to a screen capture of your results (the table and the Cumulative Win Proportion graph). Based on the 1000 simulated repetitions of playing this game, what is your estimate for the probability of winning the game with the “switch” (i.e, change) strategy?
(l) Does one strategy appear to be better than the other? If so, by a lot or just a little? Justify your answers.
(m) The probability of winning with the “switch” strategy can be shown mathematically to be 2/3. (One way to see this is to recognize that with the “switch” strategy, you only lose when you had picked the correct door in the first place.) Explain what it means to say that the probability of winning equals 2/3.[Hint: Recall our earlier definition of probability. I want an interpretation of this number, not simply an evaluation statement of whether you think it is large or small.]
(n) Suppose that you watch the game show over many years and fi nd that Door 1 hides the
car 50% of the time, Door 2 has the car 40% of the time, and Door 3 has the car 10% of
the time. What then is your optimal strategy? In other words, which door should you pick
initially, and then should you stay or switch? What is your probability of winning with the
optimal strategy? Explain.
