Below are the results and my conclusions from a problem I posed in Twitter yesterday about gambling.
Below is my tweet with the results of a poll
Sunday problem: Rich man offers this game: start with $10 and toss fair coin. If heads +$2, if tails -$1. After 100 tosses, if $10 grows to more than $40 you get 100x your net worth. If not, you lose your net worth. Will you play.? Elaborations welcomed.
— Michael Harris (@mikeharrisNY) April 8, 2018
To start with, I would like to thank all those who participated.
What I already knew and confirmed once again:
A large percentage of speculators do not understand expectation
In fact, many confuse the average with the expectation. For this game, the expectation is calculated as follows:
E = $2 x 0.5 – $1 x 0.5 = $0.5
The expectation, also known as the mean of the distribution, is equal to the average of a sample of trades only at the limit of sufficient samples (Papoulis, A. p. 138). If the sample is not sufficient, then the average may diverge considerably from the expectation. This is discussed is some more detail in my book Fooled By Technical Analysis, Chapter 2.
In gambling, there is rarely a sufficient sample to allow realizing the expectation, save the fact that in many games the expectation is negative to start with. In this particular game, 100 tosses of a coin may not be enough to realize the expectation. Some gamblers will get more than the expectation and some less. Below is the expectation path (actually the average before convergence) in the time domain for an unlucky gambler for just one sequence of coin tosses:
It may be seen that convergence to 0.5 is slow and by 100 tosses the average is below $0.4. Some gamblers will not make it to required terminal wealth in the game to get 100 times their net worth from the rich individual who is giving money away for some unknown reason.
Many of my followers are probably of young age
I arrived at this conclusion from the majority vote about accepting the gamble. This would be the decision of younger people with lower net worth who feel they can gamble and if they lose there is enough time to recover. Not many above 40 – 50 years old would take this gamble knowing that there is about 10% probability of losing everything in possession. Most rich people would probably not take the offer.
Below is a graph of 1000 random equity curves from this game.
About 8% – 10% of those equity curves finish the game below $40. This number can also be calculated from the binomial distribution but I like simulations more for this type of games.
If you have any questions or comments, happy to connect on Twitter: @mikeharrisNY
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