Te payout scheme:


• Match all 3: Win $10 • Match 2 of 3: Win $3 • Match 0 or 1 of 3: No prize


Everyone (including me) played ten rounds. If you


lost all of your money, you were out and couldn’t play any more. Te MC kept track of the winning draws on a whiteboard, so that we could reference them later. Aſter all ten rounds, I asked everyone to stand up. Ten I had them sit down if they:


• Were out of money, • Had less than $5 leſt, • Had less than $10 leſt, • Had less than $20 leſt, • Had less than $30 leſt.


When only a couple of people were leſt standing, we


compared our winnings. I did better than all but one player by playing the same seven tickets each round:


1-2-3, 1-4-7, 1-5-6, 2-5-7, 2-4-6, 3-4-5, 3-6-7. I put my seven tickets on the document camera and


asked everyone to figure out what I won or lost in each round. Tey were surprised to find that not only did I win money overall, I never lost money. In each round, either I won the jackpot (up $3 aſter buying seven tickets and winning $10), or I won exactly three of the “match 2-out-of-3” prizes (up $2 aſter buying seven tickets and winning $9). Some other players might have had bigger indi-


vidual rounds (for example winning the jackpot and one or more 2-out-of-3 prizes). But no one consis- tently came out ahead in every single round, so they wanted to know how I had done it. I promised that we would come back to that question before the end of the session.


Mathematics: Expected Value At this point, we took about 15 minutes to talk about


the odds of winning a lottery and how they are calcu- lated. Te basic rules of probability tell us that


P(event) = (# ways that event can happen) / (total # of possible outcomes).


I asked them to calculate the probability of a single


ticket hitting the jackpot in our 7-ball lottery. Some of the secondary teachers used combinatorics to calculate 7-choose-3. But the numbers are small enough that working systematically, it can be seen that:


P(jackpot) = 1/35. Te probability of a single ticket winning the


2-out-of-3 prize is harder to calculate, since you have to consider all of the possible pairs in the set of three cards (there are 3 pairs), and you have to remember that winning the 2-out-of-3 prize means that you didn’t win the jackpot. Teachers worked for a while on this question, until someone was able to convince the group that


P(2-out-of-3) = 12/35. We then defined the mathematical term “expected


value” (EV), which is really more like a weighted average value:


EV = (Probability of Event #1)(Value of Event #1)


+ (Probability of Event #2)(Value of Event #2) + … + (Probability of Event #n)(Value of Event #n).


In our lottery, there are only two events that have


any nonzero value: hitting the jackpot or matching 2-out-of-3.


EV = (1/35)($10) + (12/35)($3) = $1.31. Of course, you can never actually win $1.31; any


given ticket can only win $0, $3, or $10. We see concretely that “expected value is not the value you expect.” Since the expected value is higher than the ticket


price, this lottery game is very good for players, and very bad for whoever is using it to raise money. A player may lose in the short term, but long term you expect to come out about 30% ahead.


American Institute of Mathematics · Summer/Autumn 2016 · MTCircular 09 ⊇


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