After putting up the probability distribution and showing that after playing 100 toss of coin, where each head gives you $2 and each tail takes away $1 from you, will not always give you a positive returns, I was still unsatisfied.
I went ahead and did a monte carlo simulation of the game on excel. Using RAND() function, I will get a random value between 0 and 1. I programmed it such that if I have a number bigger or equal to 0.5, it'll be heads and I'll assign a value of 1 to it. If not, it'll be tails and I'll assign a value of 0 to it.
IF( cell >= 0.5, 1, 0)
After tallying the number of heads and tails, I proceed to count the gain/loss from this 100 toss game. I did 800 simulation of such 100 coin toss, effectively throwing 80,000 coins in my computer program.
Here's the results of all 800 such games of 100 coin tosses. The numbers in each cell represents the gain/loss from playing the game:
I highlighted 2 cells in yellow because this is what I get after playing 800 such games - I've two games where I have negative returns! The probability is higher than the initial theoretical calculation. The theoretical probability of ending up having losses after 100 coin tosses is 0.043686%. Based on this monte carlo simulation, I have 2/800 or 0.25% of ending up not being profitable.
I calculated the average score of all these 800 games of 100 coin tosses, and it turns out to be $49.43875, which is very close to the expected value calculated of $50. Just for interest sake, the mode (the sum that occurred most often) of all 800 such games is $53 and the median is $50.
There! The statement that after 100 coin tosses, you always end up being profitable is not true. Combining the new found skills in excel simulation with real life examples is really really interesting LOL