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Black Boxing · Jun 26, 07:53 PM

There is this game I’ve been thinking about; it’s for gamblers.

An input slot accepts spare change and bills of any denomination (US currency only). There is an output bin where your money comes back, multiplied by a non-negative real number and rounded to the nearest penny. Your winnings are returned in the smallest number of coins and bills possible, but on your right is a money-changer with an infinite supply.

On the front of the machine is a picture of a p.d.f. It’s the distribution from which the winnings multiplier, a random number, is drawn. An attendent stands at the side of the game to answer any questions. His answers on the p.d.f. (values, moments, integrals, etc.) are always accurate.

It’s Sunday morning, and after a night of blackjack you’ve got three twenties burning a hole in your pocket. Your flight leaves in twelve hours.

  1. How do you decide you might like to play?
  2. How do you plan to maximize your winnings?

I apologize, but this stuff’s got my attention nowadays.

* * *

  1. since i only took up to calc 2, what does the graph of a pdf look like? — Dave    Jun 28, 02:02 AM    #

  2. Clearly you want to go pennies. With a penny you get “1” for any p.d.f. value from .5 to 1. Where as for a 10$ bill you would get between 5$ and 10$ back for such values. Pennies use the rounding rule to maximum advantage.

    Decision to play: you just need to make sure that the area under the p.d.f. above 1 is greater than the area under the p.d.f. below 0.5

    Viola! Your 12 hours are productively spent pushing pennies. — Chris    Jun 28, 10:49 AM    #

  3. Dave: Any function f(x) on the real numbers can be a PDF as long as two conditions are met. First, f(x) >= 0 and f(x) < 1 must hold for all x. Second, the area under f(x) must be unity. More details than you can shake a stick at: Mathworld.

    Addiionally, I have specified that f(x) = 0 for x < 0).

    Chris: I like your solution; it shows how to profit from a loser distribution (i.e. one with an expected multiplier less than unity). When the multiplier’s mean is greater than one you may want to increase your bet some. — Tom    Jun 28, 03:55 PM    #

  4. For the interested: Chris pointed out an error in his solution, which does not account for the rounding down from 1.5 to 1.0. — Tom    Jun 29, 10:39 PM    #

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