Thesis Essays - Expected Value, Coin Flipping, Probability Theory

Thesis: In 1713 Nicolas Bernoulli proposed the following problem/paradox. It is still relatively unknown and unresearched today. You are playing a lottery game and are given a coin. The game ends when you flip tails. If you flip heads you can flip the coin again. After the game ends you are given 2k dollars where k is the amount of heads you had (e.g. flipping heads, heads, tails gets you 4 dollars). The questions is what is the expected value of the game? I will use different techniques to prove the expected value diverges and is infinite. Simulation to collect data ("Survey"): I used Mathematica 10.0 to program simulations for the game. I programmed more than 106 simulations for the data making the chance of an error greater than 1% very unlikely. A summary of the code I made follows: I generated a pseudo random 0 or 1, next I made a loop checking if it was 0 or 1. The loop repeats if the number is 1 and ends if the number is 0. Finally it counts the number of 0's and outputs 2^(number of zeros). Next I made and array with as many repetitions of the simulations as I want. Next I prepared to make a graph of the cumulative average. I added one by one the simulations then divided each with 1,2,3,4 until it was finished. Finally I called for the result to be plotted and obtained graphs. (I can include the program as an email attachment if needed) Secondary Data: Not much research has been done on this problem and remains relatively unknown in mathematics. A few sites with information on the problem follow: http://mathworld.wolfram.com/SaintPetersburgParadox.html http://statistics.about.com/od/Applications/a/What-Is-The-St-Petersburg-Paradox.htm http://www.policonomics.com/saint-petersburg-paradox/ How to investigate data: I will calculate the accumulating average. If the accumulating average continues to increase after many repetitions of the experiment then I can assume the average diverges, that is, tends toward infinity. Since the expected value is just the average value you will obtain after playing the game a giant amount of times then a diverging mean will mean my thesis is correct. There is essentially 2 independent variables, the random outcome (heads or tails), and the number of times played. These give rise to 2 dependent variables being the outcome of the game and the average outcome.