Expected Value and Analyzing Decisions, part 1

As an A.P. Statstics reader, I am excited to see the increased emphasis on statistics and probability in the Common Core standards.  Ever better, the standards specifically ask our students to be able to reach conclusions based on data:

CCSS.Math.Content.HSS-MD.B.5 (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

CCSS.Math.Content.HSS-MD.B.7 (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

This is a great start, and requires that we move beyond just flipping coins and drawing beads from bags.  We need to get our students writing about what they see, and provide strategies for developing clear, statistical thinking.

In this first post, we’ll look at a famous game show, and examine possible decisions.  Next, the game of Monopoly will provide a more complex argument in expected value.  Finally,  in a third post, we’ll look at past Advanced Placement Statistics items and adapt them for use in non-AP clasrooms.

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The game show “Let’s Make a Deal” provides a simple example in decision-making based on values and probability.  At the end of the show, contestants who won prizes during the show are asked if they would like to give up their prizes for a chance at a bigger prize by choosing one of three doors:

Should a player give up their prizes?  How much does the “big deal” need to be worth in order for a player to be tempted to give up what they won?  How about the other two doors…do they matter?  Show the first video to your class, and let them debate whether they would go for it, and take a class vote.

In the next stage, we can show what all 3 doors have behind them, and begin to consider the game as a whole:

Does this new information change any decisions?  How willing are you to risk your prizes if you know there are some other nice prizes to be won, which may or may not have the same value of what you already have?  The slides below let you walk through a discussion with your class.  Use dry erase boards or even Google Forms to have students share their ideas.

There are a number of approaches which can be taken here.  Hopefully your students develop one of these on their own, or have creative, new ideas.

• In this example, 1/3 of the doors hold the Big Deal, but another door has a prize of essentially equal value, the “medium” prize.  It may be difficult for us to predict the value of the middle prize, but it seems plusible that 2/3 of the doors will have value at least equal to what we are giving up.
• In the worst-case scnenario, the non “Big Deal” doors have minimal value to us, essentially zero.  So, we have 1/3 chance of selecting the Big Deal, and 2/3 chance of winning nothing.  We can compute the expected value:

• In the video above, we have some insight into the three doors on the show, and there always seems to be a “middle” prize and a “small” prize.  We can compute the expected value based on this information:

In any case, it seems reasonable for us to consider giving up the $7,000, but next we can think about the limits of our decisions.  What is the most you would give up to go for the Big Deal?  Can students create a general rule for making the decision?

And finally, a few parting shots for discssion:

• It’s easy to think of the prizes as a cash value only, but does the prize you are giving up matter?  What if you always wanted to go to Paris?  Does that add to the value?  Or are there prizes you would never give up?
• Expected value gives a nice summary of what we should expect to see in the long run, after many, many trials.  But on Let’s Make a Deal, you only have one shot…the chance of a lifetime.  Does this change your approach to the decision-making?

Would enjoy hearing your class experiences in sharing the Let’s Make a Deal examples.  Stay tuned for the next post on Monopoly.