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Luck in games: Can you do the math?

September 30, 2021

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When people, particularly gamblers, think about odds and probability, they usually only remember their recent experiences to determine if they have been lucky or unlucky.

Gamblers seem to be guilty of doing this more often when they sense that they’ve been unlucky. They are also prone to perceive the information in such a way to back up that idea of misfortune — or even foul play. 

Disclaimer: The following are my (Chasetheriver) observations and rules of thumb from my experience as a gambler, poker room manager, and amateur math enthusiast. If anyone wants to challenge the odds, I will be happy to listen.

Reviewing data is a valid way to check for fairness, but the most reliable way to investigate fairness is to follow a more methodical approach.

  • Try to predict what should happen, and try to assign values to how often each outcome is expected before the trial happens.
  • Make sure you run a large enough trial to make the results significant.
  • Encourage others to run identical trials to gather more data and have peer review.

Using data from what you already witnessed can be misleading. Any series of numbers will easily have a pattern or an apparent bias of some sort. Picking one incident from a game, whether it be poker or another game, is not normally enough proper evidence to prove anything. One session, or tournament, a whole week, or maybe a month is not enough.

Assuming your recent experience is concrete proof of everything is called inductive reasoning.

From its Wikipedia page:

A generalization (more accurately, an inductive generalization) proceeds from a premise about a sample to a conclusion about the population The observation obtained from this sample is projected onto the broader population.

For example, say there are 20 balls—either black or white—in an urn. To estimate their respective numbers, you draw a sample of four balls and find that three are black and one is white. An inductive generalization would be that there are 15 black and 5 white balls in the urn.

How much the premises support the conclusion depends upon 

(1) the number in the sample group, 

(2) the number in the population, and 

(3) the degree to which the sample represents the population (which may be achieved by taking a random sample). 

The hasty generalization and the biased sample are generalization fallacies.

Try an experiment.

Imagine you are about to roll a six-sided die six times. Take a moment to note what you would answer to these questions. (Answers at the end.)

  1. What are the chances all six numbers will come up?
  2. What are the chances all six rolls will be the same number?
  3. What are the chances that if you guess a number before each roll you will guess wrong all 6 times?
  4. What are the chances a 4 never comes up?
  5. For Q3 and Q4, which outcome is most likely?

Even an experienced card or dice player might be surprised with the odds, but most will get Q5 right without knowing the exact numbers.

How does this help to judge probability?

How often do you see a whole series of random events and not even try to imagine the odds of what you just saw happening?

It is common to look back with hindsight, choose a starting point, and want to calculate the chances even though you now know the outcome.

Another test is to imagine the last three-card flop you saw (as an observer with no hole cards) in a community card game like Hold’em or Omaha. How likely were those cards to be dealt?

Naming three cards drawn in any order from a pack of 52 (remember: you personally don’t have hole cards) is one in 22,100. 

Did you pause to think how unlikely that was, or just accept them as standard? Pick up a pack of cards at home, or choose three cards in your mind. 

Guess what? Still 22,100 to one!

When you last got exasperated because you missed another draw, and worked out it was the fourth or fifth time in a row, and that must be hundreds to one against, how does that compare to 22,100 to one?

Now, think back to a time when you filled two or three draws in a row and won some pots.

An inductive observer would note you “always hit your draws” in that sample, or even suspect you were manipulating the outcome.

Bottom line

Good or bad luck can be tricky to quantify. 

Winning the lottery is obviously way above expectation. Losing for a short period of time in a game where you expect to win is still very common and completely normal. 

This does not mean you don’t have to investigate the mechanics of the game, or ask others for their thoughts, but try to keep a perspective on what you witness and not jump to conclusions based on feeling and short term alone.

Answers:

  1. 1 in 65
  2. 1 in 7,776. If you said 1 in 46,656, it is because you did not take into account the first roll can be any number and only the next 5 rolls are 1 in 6.
  3. About 33.5%, 1 in 3
  4. About 33.5%, 1 in 3
  5. Exactly the same. It doesn’t matter if you pick a number at the start, or change your mind after watching some outcomes.