Quiz

  • Goals

    • Learn a basic interpretation of the term probability.
    • To use simulation to investigate the underlying properties of a random process.

  • Background

  • A popular television game show (Let's Make a Deal from the 1960s and 1970s, revived in 2009) featured a new car hidden behind one of three doors, selected at random by the producers. Behind the other two doors were less appealing prizes (e.g., goats!). When a contestant played the game, they were asked to pick one of the three doors. If the contestant picked the correct door, they won the car!

    However, there was a twist! Once a contestant picked a door, instead of showing the prize behind that door, Monty would reveal a goat behind one of the other two doors. Then he gave the contestant a choice - stick with their original door choice or switch to the remaining unopened door.

    For example, suppose you pick Door 1 and then Monty shows you that there is a goat behind Door 3. Now you have to decide, do you want to move forward with Door 1 or switch to Door 2, or does it not matter?

    Assuming there is no set pattern to where the game show puts the car initially, this game is an example of a random process: Although the outcome for an individual game is not known in advance, we expect to see a very predictable pattern in the results if you play this game many, many times.

    		 Monty Hall hosting Let

    So let's play the game!

  • Computer Simulation

    Simulating a random process

  • To determine whether one strategy (staying with your first door choice or switching to the other door) is better than the other, you need to play the game many times with each strategy to see whether you find any meaningful difference in how often you win with each strategy. The applet to the right will allow you to select a strategy and play the game with the computer playing the role of host.

    • 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 these 10 plays where you won the car in the table below.

    TAIQ: Do you think everyone in class will have the same 10 outcomes?

    • 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 below.

    • Now uncheck the Animate box and 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.
  • Rows
  • Rows

  • Probability

    Definition
  • 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.

  • Computer Simulation

  • 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 conjecture.

     

     

     

     

     

     

     

     

     

     

     

    • 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 box.
    • Press the Go button.

    When you create output like this, we will often want you to save your results.

    Options for Screen Captures.

    Create a screen capture of your results (the table and the Cumulative Win Proportion graph).
  • Browse Files
    Drag and drop files here
    Choose a file
    Cancelof

  • 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.)

  • stopsign
    The previous question is pretty important, ask the instructor of TA if you are not sure.

  • Application

  • (j) In this course we will use the "long-run proportion" definition of probability to determine and make sense of probabilities. To that end, here is some practice at re-phrasing probability statements using the long-run proportion definition.

    For each statement, be sure to identify:

    i. what random process is being repeated over and over again (what are the identical conditions, e.g., picking a door at random) and

    ii. what proportion is being calculated (e.g,. proportion of wins).

    Your answer should not include the words “probability,” “chance,” "odds," or “likelihood" or any other synonyms of "probability."

  • Summary of technology skills learned/practiced in this lab

    • Using the online lab instructions, running a javascript applet, and putting answers into a Word file
    • Making screen captures of applet output

  • Summary of statistical skills learned in this lab:

    • Interpreting probability as a long-run proportion.
    • Simulating outcomes of a random process in order to investigate the underlying properties of that process.
  • Should be Empty: