Re: Weekend Puzzle

Andor wrote:

Jerry Avins wrote:


This is an old puzzle, due to Daniel Bernoulli. It has a name I'll
withhold for a while. One plays this game: a fair coin is tossed until
it comes up tails, the payout being 2^(n-1), where n is the number of
tosses. What is the expected value of winnings assuming many games are
played?


It's not bounded. However, in a certain sense, it is equal to -1/2.
Interested?


What entry fee would make it a fair game?


Since the expected value is infinite, any price is too low to make it
fair.


What would you pay?


That depends. How many times may I play? For a one off chance, this is
a complex psychological problem (sort of like the Prisoner's Dilemma).
If I can play repeatedly, the Law of Large Numbers should give a good
hint.

Try it and see. If a $64 payout would break the bank, how much would you
pay? $1,048,576? $1,099,511,627,776?

This reminds me of another probability puzzle (due to Mandelbrot, if I
remember correctly). Imagine an archer standing on the y axis and
facing an infinitely long wall along the x axis (Mandelbrot is a
mathematician:-). The archer is spun around blindfolded, which
effectively points him in a direction uniformly distributed between
-90° and +90° (while still facing the x axis from his position). He
then shoots his arrow. Two questions:
1. What is the expected x coordinate of the point of entry of the arrow
(remembering that the archer is standing on the y axis)?
2. What is the expected length of the flight path of the arrow?

(Hint: the two questions do not have the same answer).

Maybe next week.

Jerry
--
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
.

Relevant Pages

  • Re: With all the pricing changes...
    ... Comps are not down... ... The Blackjack 6-5 payout is designed for people who don't understand ... the game, playing a version of the game no one who DOES understand the ... WANT the recreational player to believe is THE game to play. ...
    (rec.arts.disney.parks)
  • Re: Weekend Puzzle
    ... One plays this game: a fair coin is tossed until ... How many times may I play? ... What is the expected x coordinate of the point of entry of the arrow ...
    (comp.dsp)
  • Re: Weekend Puzzle
    ... One plays this game: a fair coin is tossed until ... Observe that the probability of n tosses is 1/2^n, and that the payout ... the expected payout for any ...
    (comp.dsp)
  • Re: Weekend puzzle.
    ... One plays this game: a fair coin is tossed, ... where n is the number of tosses before "tails" is ... the payout in that case is 2^n. ...
    (comp.dsp)
  • Weekend Puzzle
    ... One plays this game: a fair coin is tossed until ... Observe that the probability of n tosses is 1/2^n, and that the payout ... the expected payout for any ...
    (comp.dsp)