Randomness is too Important to Leave to ChancePlease email NicholasKarlson@gmail.com to report errata or to make suggestions.EXAMPLEMathematics can be used
to model economic ideas. Math is also good at modeling randomness. To
see how simulation can replace math in economic modeling a good place
to start is to see how it can replace math in modeling randomness.
Consider a simple dice game called dgame. There are two players in
dgame (player1 and player2). Player2 starts by rolling two six-sided
dice. If the dice sum to 7 or 11, player2 wins and the game is over.
If the sum of the dice is 2,3, or 12 player1 wins and the game is over.
If the dice sum to the number x ( x being 4,5,6,8,9, or 10) then
player2 keeps on rolling the dice until either the dice sum to 7
(player1 wins) or the dice sum to x (player2 wins).This
is an amusing game of chance that you could play with a friend. Which
player has the advantage? If you were playing an even simpler game
(flipping a coin to see who wins) the obvious answer would be that
neither player has an advantage since a "fair" coin would be "heads" 50%
of the time on average (or the probability of "heads" is 0.5). This is
a very basic math model of flipping a coin. It is a model, because it
is trying to describe reality but it is not reality (suppose the coin
for your game is weighted in a weird way so if you flipped it 1000 times
it came up heads 905 times?). Returning to dgame, it turns out that
mathematical probability theory can be used to model the chance of
player1 winning. The mathematical model's answer is 251/495. The
important assumption to get this answer is that the dice are fair and
any given roll has no effect on any other roll.Instead
of using math to derive 251/495 why not write a computer program to
model the chance of player1 winning? Would either method be more valid
than the other? Would either method be more practical in modeling
reality?Try a program that implements the above at a website hosted by MIT: http://scratch.mit.edu/projects/nkarlson/1786438 Original material on this website copyrighted 2011, 2014, 2018 please contact NicholasKarlson@gmail.com for details. |