Expected Values

I feel like the idea of “expected values” may be the most important practical concept in probability – and yet, sadly, I find that I completely don't have time to discuss it in any of the math classes that I teach. (Nor is it part of the primary “story” of any of the classes, even statistics). Ever semester I re-visit my schedule and try to find a day to cram it into, and realize again that I cannot.

I've found that probability is enormously alien to a surprising number of students. (Just last week I had students in a basic math class fairly howling at the thought that they might be expected to be familiar with standard dice or a deck of cards). Therefore, I find that I actually have to motivate these discussions with an actual physical game, of the most basic simplicity. If I did cover expected values, here's the rudimentary demonstration I'd use:

The Game: Roll one die.
Player A wins $10 if die rolls {1}.
Player B wins $1 if die rolls {2, 3, 4, 5, 6}
Calculate probabilities (P(A) = 1/6, P(B) = 5/6).

Let a student pick A or B to play, roll die 12 times (say), keep tally of money won on board (use I's & X's). Likely player A wins more money.

Expected Value: The “average” amount you win on each roll.
E = X*P (X = prize if you win; P = probability to win)
Calculate expected values.

Ex.: Poker situation.
If you bet $4K, then you have 20% chance to win $30K. Bet or fold? (A: You should bet. E = $30K * 20% = $6K. If you do this 5 times, pay $20K, expect to win once for $30K, profit $10K)

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