A man appears from nowhere and tell you that he knows your total worth is X amount of money. He proposes to you a game of tossing a fair coin (independently confirmed to be fair) just twice and he will pay you X amount, equal to your total worth, each time heads show up. However, you will lose half of that, or X/2, each time tails show up.
You think for a moment. Something does not sound right here you suppose. Then you remember that boring course in Probability Theory and that there are four possible outcomes to this game:
{heads, heads} , {heads, tails} , {tails, heads} , {tails , tails}
Each outcome above, actually called an element of the Probability Space in the theory, has probability equal to 1/4 because each tossing is independent. You then calculate your expected wins or losses from each possible outcome:
{heads, heads}: w1 = X + X = 2X
{heads, tails}: w2 = X – X/2 = X/2
{tails, heads}: w3 = -X/2 + X = X/2
{tails , tails}: w4 = -X/2 – X/2 = -X
“This is it”, you think, “This is the chance to make it big”. The probability of winning is 3/4 and the probability of losing everything is just 1/4. Thus, if you win, you can make either 2X or X/2 and if you lose, you lose all your wealth.
While you are thinking that this may be a good deal, a formula you learned once for calculating the expected value E comes to mind. You apply it to the numbers above:
E = 1/4 x 2X + 1/4 x X/2 +1/4 x X/2 +1/4 x (-X) = X/2
In other words, the expected value E is the sum of all possible values times their probability, given that the probabilities sum to 1.
The question is, are you going to play this game? You have 25% chance of winning twice your total worth, 50% chance of winning half your total worth and 25% losing it all. The expected value is also positive and equal to X/2.
While you are thinking about it, your friend comes in and glances over the calculations. “Go for it”, he says. “The expectancy of this game is positive. This is your chance of becoming rich at the expense of that fool who is throwing money away”.
You are just about to sign the contract but suddenly you remembered that your friend flanked Probability and Stochastic Processes when you were both engineering students and then he switched to Marketing. “Hmm….”, you are thinking. “…better think twice about it…where is the catch?”.
To make a long story short, the catch is that the expectation, expectancy, or whatever you may call it, formula applies only in an n-time repetition of the game, with n being sufficiently large [1]. In other words, you will make X/2 on the average per game only if the times you play the game is sufficiently large. This is probability theory but also common sense.
Another defense of common sense
One of the greatest powers in this world is common sense. It is so powerful that some people were able to achieve great things by using this power even when they lacked formal education.
It should be common sense that the 25% probability to lose it all is just too high to play the game. It should also be common sense that you need to play this game many times to realize X/2 per game since there is a finite, although small, probability of {tails, tails} showing several times in a row. Would you be willing to risk all your worth in such a game although the longer-term expectancy is positive?
I would not, if I was told that I could play the game only once but also if the bet was equal to my total worth. Relying on a positive expectancy for a single trial is a huge mistake that is actually caused by misunderstanding/misapplication of probability and statistics. In addition, to be able to stay in the game in the long term and eventually realize the positive expectancy requires smaller bets, actually very small, each time the game is played.
This is related, of course, to traders and trading systems. Even in the face of a positive high expectancy risk of ruin is possible depending on the size of the bets and sequences of winners and losers. Optimal betting systems, like %Kelly, should be avoided. Trading is the art and science of surviving adverse price moves while realizing a longer-term positive expectation using carefully designed position and risk management.
References:
[1] A. Papoulis, Probability, Random Variables and Stochastic Processes, McGraw-Hill, Inc. 1965. p. 138
