V

#### virtualadepts

In a fair game of coin toss, where the odds reach an equilibrium of

50/50 chain reactions do occur. This can be explained using martingale

probability theory, but in simpler terms it only shows an example of

how order emerges out of chaos.

Example: One player has 3 pennies, and another player has only 1 penny.

A fair coin is tossed every round to determine if a penny is won or

lost for either player. The odds are 3/4 that player A (Who begins with

3 pennies) will win the game. This is entirely different than the

martingale betting strategy, because only 1 penny is bet for each round

of the game.

Because there are 3 ways player A may win, and only one way player B

can win, player A has a concrete advantage. Player B, only wins in the

event that the coin is tossed in his favor 3 times in a row, while

player A can win on the first throw. Or he can win after losing the

first coin toss, or he can win after losing the second coin toss. So

the odds are 75% that he (or she) will win in this game.

Upon further analysis it is possible to calculate the average number of

coin flips before player A is likely to win. The equation k(n-k) works

for perfectly fair games according to martingale probability theory to

solve this problem. In this case 3(4-3) solves the problem, so on

average it takes 3 coin flips for player A to win.

To show that chain reactions occur you only have to move from the

probability of winning the first game, and multiply it by the

probabilities of winning the following games. For example, if 3 pennies

are used to play this game in an attempt to win one penny, the odds are

3/4. And once that penny is collected there is now a 4/5th chance of

winning another penny.

So statistics tells us that there is a (3/4) * (4/5) * (5/6) * (7/8) *

(8/9) * (9/10) = 30% chance of the 3 pennies growing into a pile of 10.

But in repeatable tests you will find that on average there is not a

net win or loss in this game. If there is a 75% chance of winning 1

penny, and a 25% chance of losing 3. The two odds cancel each other

out, to create an equilibrium in 50/50 games.

And at the same time we can see that despite the fact that the initial

value of coins reaches an equilibrium when the pattern is extended to

any length, we can show a concrete advantage to beginning with 3

pennies, instead of beginning with one.

In the last example player A had a 30% chance of winning 7 pennies, and

totaling 10 in all. If we started with only one penny then player A

would just have to total 8 pennies in order to earn 7. So lets look at

the math:

(1/2) * (2/3) * (3/4) * (4/5) * (5/6) * (6/7) * (7/8) = 12.5%

So we can cleary see that even though winning 7 pennies has the same

expected value as losing 1 penny. Outside of repeatable tests the odds

of earning 7 pennies is clearly higher if you begin with 3.

I can also explain the laws of nature with these prinicples. If we

look at the equation for gravity on earth, which accelerates at 9.8 m/s

we can derive an acceptable answer from the earlier equations. The

gravity equation I am using is sqrt(2*n/9.8).

In this example we are dropping a ball from 4.9 meters, and you can see

it takes one second to land.

t = sqrt( ( 2(4.9 m) ) / ( 9.8 m/s^2 ) ) = 1 s

So here is my gravity theory. We are using the quadratic formula to

solve: 2*n/9.8 = k(n-k) , for k. (The formula k(n-k) finds the average

number of coin flips).

k=(1/14) (7n +- sqrt(49 n^2 - 40 n)).

So now an example...

We are dropping a ball from 10 meters above the ground. So we plug 10

meters into n to solve for k.

k=(1/14) (7n +- sqrt(49 n^2 - 40 n))

k=9.791574237

My question to calculate the average number of coin flips in my game is

k(n-k), so we plug in k & n:

k*(10-k) = 2.040816327 = average number of coin flips

Now we take the square root of the average number of flips to get the

actual time it takes to land:

sqrt(avg flips) = 1.428571429 = number of seconds to land.

Now finally to factor in a problem with my equation we say that if k is

9.791574327, that means our large gravity pile is that many pennies.

And our small gravity pile is exactly 0.208425673 pennies!

Now for the source code. You can actually prove everything I have

written by running a few simple test cases. In the program when you

set the initial beans to 5, and set 1, 2, 3, 4, or 5 beans as your

goal, the output should look like this:

5:1 = 83.5%

5:2 = 71.6%

5:3 = 62.6%

5:4 = 56.1%

5:5 = 50%

But if you only play with 5 beans every time and only go after 1 bean

with those 5 each game, then your output will look like this;

5:1 = 83.5%

5:1 X 5:1 = 69%

5:1 X 5:1 X 5:1 = 58%

5:1 X 5:1 X 5:1 X 5:1 = 48%

5:1 X 5:1 X 5:1 X 5:1 X 5:1 = 40%

So there is all the proof you need. Which experiment would you rather

play?

Another experiment you can try:

Modify the program to run 10,000,000. Starting with 3 beans each time

with a target of 5000. You will win 5000 beans 273 times, for winnings

of 1,365,000 beans in total. And you would lose 3 beans 443323 times

for a loss of 1,329,969 beans. So you ended up 35,031 beans ahead.

Try changing the seed and you will still be ahead in the long run.

Even if you play 100 million games you will still be ahead.

*/

#include <stdio.h>

#include <stdlib.h>

main ()

{

double r;

long int M;

double x;

int y;

int z;

int count;

int seed = 10000;

printf("Enter seend for RNG: ");

scanf("%d", &seed);

srand (seed);

M = 2;

int score = 0;

//Score keeps track of the number of beans won every game

int games = 0;

// games keeps track of the number of games we have played before

//losing all of the beans, which is equal to score.

int beans1 = 0;

// Initial value set to zero and defined within the loop

int wins = 0;

int lost = 0;

int quit = 0;

int init = 0;

int rounds = 0;

int live = 0;

printf ("Initial Beans: ");

scanf ("%d", &init);

printf ("Stop after winning X number of beans: ");

scanf ("%d", &quit);

printf ("Number of rounds: ");

scanf("%d", &rounds);

printf("Show live output (1 or 0): ");

scanf("%d", &live);

for (int cnt = 0; cnt < rounds; cnt++)

{

// We play up to (int) rounds

int count = 0;

beans1 = init + score;

// Beans gets defined here, as starting with 3 beans

// and having a 0 bonus score (It changes as you

// win more beans per round)

int beans2 = 1;

// The program attempts to win just one

// bean for every game.

while (beans1 != 0 && beans2 != 0)

// The battle begins

{

r = ((double) rand () / ((double) (RAND_MAX) + (double) (1)));

x = (r * M);

y = (int) x;

z = y + 1;

// A coin is flipped and is either 1 or 2 in value

if (z == 1)

{

// Heads wins.

beans1++;

// Beans1 gains one bean from Beans2

beans2--;

}

if (z == 2)

{

// Tails loses

beans1--;

// Beans2 gains one bean from Beans1

beans2++;

}

count++;

// We keep track of the number of rounds in the battle

}

if (beans1 > score + init)

{

// If beans1 is greater than the initial value

// of beans plus the total number of beans

// that have been won so far in this game, then

// the score goes up, and we go on to the next

// game. We check this at the end of every game.

score++;

games++;

}

if (beans1 <= 0)

{

//If beans1 has lost the game and doesn't

//have anymore beans then we know the

//game is over, so we reset score, and reset

//games.

if(live==1){

printf ("Lost at: %d beans , %d games.\n", score + init, games);

}

// And we print out the total number of

// games played on this trial and show the

// total score plus the initial value of beans.

lost++;

score = 0;

games = 0;

}

if (score >= quit)

{

wins++;.

if(live==1){

printf ("Won at: %d beans , %d games.\n", score + init, games);

}

beans1 == 0;

score = 0;

games = 0;

}

}

printf ("Total Won: %d/%d\n", wins, wins + lost);

printf ("Net win: %d beans.\n",(wins*quit)-( lost*init ) );

}