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What is the Chance of Losing All My Money Short-Term Trading Stock? According to a Statistical Concept: the Gambler's Ruin Problem

Ideally, we want to be able to "buy low, sell high" when doing short-term stock trading.

However, there is a chance we will "buy high, sell low" and lose money.

According to a statistical concept, the gambler's ruin problem, tradning stock can be very dangerous and we may lose all our money trading.

The Gambler’s Ruin Problem

The gambler's ruin problem can be described as two gamblers, A and B, gamble 1 dollar each time. A has x dollars, and B has n - x dollars. They have total of n dollars and they gamble until one of them loses all his or her money.

Let the probability that gambler A wins a game be p, then the probability of A losing a game is q. p + q = 1, since A either wins or loses a game.

If  p does not equal to q, The probability of A wining all the money is
If  p equals to q, The probability of A wining all the money is

Analysis of 3 scenarios of trading

Assume each trade is either wining or losing 1 dollar for calculation purposes.

scenario 1: short-term trading with the probability of making money in each trade being 50%.

Let gambler A be a short-term trader and every time A conducts a trade, A's probability of making money in each trade is 50%.

Assume A will continue trading until he or she loses all his or her money, the probability of A losing all the money is 100%! 

This results from n being very large as in the equation
A's probability of wining everybody's money is 0, and on the other side of this equation, A will definitaly lose all A's money.

scenario 2: short-term trading with the probability of making money in each trade being 49%.

If A's probability of wining in each trade is 49%, A will lose all the money more quickly. Even if A is not trading against a lot of people that together have almost infinite amount of money.

Below is a table of  A's probabilities of losing all A's money with respect to how much money his or her trading competitors have:
A's Money Other People's Money Chance of Bankruptcy
$100 $10 33.4%
$100 $50 86.7%
$100 $100 98.2%
$100 $200 99.9%

scenario 3: short-term trading with the probability of making money in each trade being 51%.

If A's probability of wining in each trade is 51%,  can A make money in the long run? Refer to the probability table below:
 
A's Money Other People's Money Chance of Bankruptcy
$1 $1,000,000 96.1%
$2 $1,000,000 92.3%
$5 $1,000,000 81.9%
$10 $1,000,000 67.0%
$20 $1,000,000 44.9%
$50 $1,000,000 13.5%
$100 $1,000,000 1.83%
$200 $1,000,000 0.03%

Conclusion

If one wants to do stock short-term trading, one needs a wining probability in each trade greater than 50% and has enough money to be able not to lose all one's money.

If one's probability of making money in each trade is 50% or lower, one will go bankrupt in the long run if one keeps trading . 

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