The Gambler’s Fallacy

 

 

From the standpoint of logic, it seems quite salient that human thought is blighted with a plethora of logical fallacies.  Fallacies ranging from the misapplication or misleading heuristics to obdurate ratiocinations that blatantly rejects the established unassailable facts of reality. The spectrum of reasoning errors as you can imagine is quite vast. Hence why there is a multitude of various logical fallacies that are commonly discussed in the subdiscipline of Philosophy known as logic. The question then becomes why precisely we as humans are so incredibly susceptible to such faulty thinking?  While humankind is far from perfect it is awe-inspiring how flawed our thinking is. The follies and pitfalls of human thought have a wide range of generally latent motives that act as the prime mover of such errors.  Not all intentional even at the subconscious level, but rather are the byproduct of the conflation or transposing different variables. Which in turn leads to faulty reasoning and the perpetuation of erroneous assumptions.

 

The extent to which logical fallacies are prevalent conjures a plethora of questions regarding their role in human thinking. If we are to accept the Darwinian theory of Evolution as true, then there is the potentiality that there is an adaptive mechanism behind our faulty thinking. While the postulates of Charles Darwin are fixated around random gene mutation, the core assumption of his theory that adaptations that persist generally have some sort of advantageous benefit. A superlative example of this is the double-edged sword of the condition of Sickle-Cell Anemia. Such paradigms of their being a blessing and a curse in embedded in genetic alterations extends beyond congenital conditions. It has been well discussed how the human brain has achieved adaptive gains through cognitive and perceptual biases. For example, I recall an Evolutionary Psychology professor stating that the reason why most people do not have photographic memory is that those who do possess such prodigious capacities for recalling information often suffer from poor comprehension skills. Such a tradeoff is certainly exemplary of the limitations of the human brain. However, this does align us with the potential precedent for extrapolating this concept to cognitive adaptation. Which makes it pertinent to question if there is an implicit utility to cognitive biases and erroneous patterns of reasoning.

 

The field of Social psychology thoroughly addressed most of the heuristics and cognitive biases that people commonly exhibit. To some extent, it is fruitless for me to delve too deep into such inquiries most would say that we already have cogent and empirical explanations for such phenomena.  Regardless I am still allured by the perplexing rabbit hole of human reasoning. How we can manipulate evidence and stretch it beyond the limits of the truth and present it as factual is truly a mind-boggling spectacle. Much of our thinking is plagued by superstitions and faltering attempts to rectify cognitive dissonance.  A lot of this inadvertent or intentional does possess a functionality, whether it is subject to evolutionary lag is another story. For example, generally racial prejudices have been perceived as an adaptative social mechanism.  If you think from a historical standpoint there is an underlying logic to such thinking. Back when humans were more nomadic and prior to permanent agricultural settlements violent conflicts over resources were much more prevalent. In that era of human history, it was important to insulate yourself from predication which entailed developing biases against individuals who were not a part of your immediate community. The consequences of not doing so were so grave that it was literally a matter of life or death.

 

However, even as human civilizations flourished and became more advanced the prejudices for people who are different than ourselves remain resolute. While the norms around general deportment shifted from outright barbarism to more civil interactions the ingrained nature of man persisted. Prejudice has acted as a quick means of ascertaining who is a member of your community and whether you need to be wary of them. It has been ingrained in humans to the point it acts almost like an automatic inference. Stereotyping almost acts as a mild hangover from more severe forms of prejudice, however, it still functions as a quick heuristic to make a snap decision. While the magnitude of such attributions has been being diminished over time, their remnants still linger. Please note that this is a value-neutral attribution regarding prejudice. I am more interested in looking at the impetus behind prejudice then passing moral judgment.

 

The logical fallacy I will be highlighting in this discourse will be the Gambler’s Fallacy which on the superficial level does not seem so faulty. However, upon a deeper assessment of the reasoning involved, it becomes quite conspicuous how flawed it is. A prime example of the wrong variable being analyzed, and the actual facilitating variable being forfeited. Analogous to the example of the adaptive basis for ethnic prejudices, the reasoning has an outmoded utility that now yields detrimental results. Both examples differ from the example of Sickle-Cell Anemia in two ways. The first being that Sickle-Cell Anemia while profoundly detrimental, there is still the benefit of immunity from Malaria.  While the potential benefits of the bias built-in to the Gamblers fallacy is outmoded if not nonexistent. The Second and most obvious difference being Sickle-Cell Anemia is a form of physiological adaptation, while the Gambler’s Fallacy is cognitive.

 

The intentions of this blog entry will not be so much to ascertain if there was ever an evolutionary basis for the Gambler’s Fallacy, but rather to provide an overview of the concept. As is evident above I will interweave commentary and speculation regarding the mechanics, origin, and other opinions pertaining to this common fallacy.

 

WHAT IS THE GAMBLER’S FALLACY?:

 

In a broad sense, the Gambler’s Fallacy can be defined as the error in logic where an individual assumes that the results of independent events can influence the probability of another random event [1]. One of the most conspicuous applications of this fallacy is as the name indicates in gambling. Oftentimes gamblers attempt to ascertain the probability of winning on erroneously derived patterns. Due to the independent nature of the events comprising the illusionary pattern, the players perceive a pattern when one does not exist.

An excellent example of this fallacy follows as below:

“..Example of Gambler’s Fallacy

Edna had rolled a 6 with the dice the last 9 consecutive times. Surely it would be highly unlikely that she would roll another 6 on the 10th time …” [1].

This example demonstrates how this fallacy assumes interconnection between events that are completely independent. In the example above, the probability of rolling a 6 remains the same on the 10th time as it did on the 1st time.  Each roll of the dice is a separate action that’s once completed cannot be continued. Clearly delineating the separation between actions. Due to such a distinction, Ludwig Von Mises, detailed in his treatise Human Action how the Gambler’s Fallacy demonstrated a confusion between Category and Instance  (pages 110-115) [2]

 

INVERSE GAMBLER’S FALLACY:

The Inverse Gambler’s Fallacy is the natural reciprocal of the Gambler’s Fallacy; however, it does operate on a very similar principle. In a similar manner, it assumes that independent variables are interconnected. This fallacy entails that an individual believes that a genuine outlier has occurred more frequently generally with more mundane results [3]

Example below:

“… the inverse gambler’s fallacy is committed by someone who enters a casino and, upon witnessing a remarkable outcome at the nearest table—says, a five-fold six in a quintuple die toss—concludes that the toss is a most likely part of a large sequence of tosses…” [3].

Again, as is demonstrated by the above example both related fallacies suffer from the fault of assuming interrelation on independent events.

 

THE PSYCHOLOGY BEHIND THE GAMBLER’S FALLACY:

The human brain often attempts to act as an equalizing force regarding human perception and reasoning. Now some may say my assertion here is somewhat flawed considering the brain acts as the total processing system for all life-related processes. Analogous to the Central processing unit embedded within a computer. However, while that is true, the human brain utilizes various short cuts in attempts to better navigate the world around us. The dizzying array of external stimuli and information the human brain only has so much bandwidth to take it all in. Considering the malleability and adaptive features of the human brain it stands to reason that our brains work to achieve a sense of symmetry in the world.  At times it imposes that continuity and symmetry upon our sensory output causing us to perceive things in a slightly inaccurate manner. In a manner, that helps us better assimilate to a vast and multitextured environment.  For example, Impressionist painters of the 19th century manipulated our brain’s propensity for rectifying ambiguity into more manageable forms. The fact that their paints appear to be cohesive and solid images from a distance but are rather unrefined and broad-brush strokes. [4] It is clear that our brains do seek to adjust and presume ambiguity to help us better act and thrive in our environment. With all the sensory information not, all of it necessarily pertinent to our day to day function.

I would postulate that the same would be true of human reasoning, as it is with human perception because often there is so much information that can weigh down that it would impede decision making. With the abundance of research related to the phenomenon of apophenia, I would assume that my inclinations are far from unique. One prevalent form of Pareidolia is defined as the tendency to see familiar figures in random objects [5]. Typically, the error made when some claim to see the image of Jesus Christ in the grooves of burnt pieces of toast.  Much like Pareidolia, the Gambler’s Fallacy is a form of apophenia. The tendency to see meaningful relationships between independent events [6]. The cognitive behavior of looking for patterns where they do not exist fits precisely the nature of the Gambler’s Fallacy.

While our brains have the propensity to seek patterns even when confronted with random events are there potential predispositions that make people less susceptible to this logical fallacy? It appears as if the variable of experience may make someone less apt to fall prey to the Gambler’s Fallacy in decision making. Prior to composing this blog entry, I reviewed a plethora of various sources pertaining to the Gambler’s Fallacy.  I recall several YouTube videos that described it as a fallacy that “unseasoned” or “amateur” Gamblers fell into. In contrast, Gambler’s that can proficiently count cards know better than to get hindered by it. Certainly in a priori / unempirical sense would imply that experience curtails the aptitude of the fallacy being committed. The research would seem to substantiate this notion. Various studies have yield results that would illustrate a lesser prevalence among more experienced discussion makers [10]. The potential application of the results can extend well beyond the faulty attributions of gambler’s futilely attempting to beat the house.

A study conducted by the University of Chicago on the behalf of the National Bureau of Economic research demonstrates this point. The study examined how the Gambler’s Fallacy impacted the decision making of  Loan Officers, Asylum Judges, and Baseball Umpires. The researchers conducting the multi-tiered study purported to have controlled for variables sufficiently avoiding confounding the error with other reasoning related errors. The results yield from the study suggests that the more experienced the decision-maker the less apt they were to commit the fallacy. Considering that the decision-makers analyzed came from three nonrelated disciplines the ability to extrapolate the results to other situations is quite high [7].  I would surmise the mechanism behind more experienced decision-makers avoid this faulty thinking stems from having more familiarity. Being more acquainted with the specific variables entailed in properly assessing a situation make it easier to have more confidence in the decision-making process.  The ability to ascertain related versus unrelated variables may take prior exposure which facilitates better comprehension. In the absence of the understanding of the relation between various variables, we are more apt to lean on superstitious reasoning.

THE LOGIC BEHIND THE GAMBLER’S FALLACY:

One of the basic and most notable faulty assumptions inferred when the Gambler’s Fallacy is committed is assuming that independent events are interconnected. Whether or not event A and event B are connected while profound influence over the calculations of the probability of both events. This is why when we try to infer meaningful patterns where none exist we are led awry. In order to properly determine the probability of when an event is likely to occur when need to understand when variables are connected and when they are independent of one another.

 

MUTUALLY EXCLUSIVE EVENTS:

Mutually exclusive events in the discipline of probability are events that cannot coincide simultaneously. Mutually Exclusive events are often times transposed with Independent events. Even though there is a clear distinction between the two [8].

Mutually exclusive events are seen as being a “disjoint”, meaning that the probability of both occurring at the same time is zero. Such a circumstance is expressed in the below equation:

Disjoint: P (A & B)= 0      [9].

The issue then becomes how do we determine what is the likelihood that either event A or event B will transpire. This inquiry is remedied by calculating the sum of either event occurring. This is what is known as the Rule of Addition and is only applicable for mutually exclusive events.  Which is expressed in the below computations:

P (A or B) = P (A)+ P (B)

P(1) = Probability of 1/8

P(2)= Probability of 1/8

P (1 or 2) = P (1) + P (2)

= 1/8 + 1/8

= 2/8, = 1/4        [10].

 

INDEPENDENT EVENTS:

From the standpoint of probability, Independent events are events that the occurrence of one event does not impact the probability of the other. Unlike Mutually exclusive events that cannot occur at the same time, Independent events can simultaneously. The rule of specific multiplication is applicable for determining the probability of independent events occurring

P (A and B) = P (A) * P (B)

Event A’s Probality= 0.6

Event B’s Probality= 0.4

0.6*0.4=0.24             [10].

 

DEPENDENT EVENTS:

Events where one event impacts the aptitude of whether or not the other event will transpire. The rule of Conditional Probability is utilized to determine the probability of dependent events.

Conditional Probability: P (B|A) – (probability of  B given A)

It should also be noted that the probability regardless of the relation of the events we can utilize the rule of general multiplication to determine the probability of the events occurring.

P (A and B) = P (A) * P (B|A) [10]

 

Ultimately the Gambler’s Fallacy involves confusing Mutually Exclusive events and Independent Events with Dependent Events.  When making a wager on the results of a spin on the roulette wheel the outcome has no connection to any previous outcome. Nor can the dial of the roulette wheel fall definitively fall on two numbers at once. If there is an extremely improbable chance that the dial falls on the same number three times in a row, this does not alter the odds of it happening a fourth time. The mistake most gamblers would make is that they would anticipate that the odds of the dial falling on that same number is significantly reduced. All because of the results of a coin toss resulted in the quarter landing tails four times in a row, the odds of it landing tails on the fifth flip of the coin is still 50 percent.

DISCUSSION:

Gambling is an activity that has long been associated with superstition and erroneous reasoning. As someone who use to be employed at a casino, I can certainly verify this from an anecdotal standpoint. Those who engage in gambling tend to confuse variables of causation, regardless of causality generally the odds are against you. It should be noted that a casino operates as a business, not a charity. Therefore the illusion of everyone having a good probability of winning big is a true fantasy.

The first formal recognition of this fallacy may have its origins on the Casino floor it actually can be seen in many other instances of decision making. It can be certainly surmised that this cognitive bias resulting from confusing variables of causality. Which is an issue that the scientific method has grappled with for a very long time. Science aims to rectify the third-variable problem through the presumably air-tight methodology.  Through the application of Occam’s Razor nullifying the validity of any postulations that cannot be quantified and subjected to the gauntlet of empirical analysis. Hence, why science is held in such high regard because of the built-in inoculation from superstitious thinking. Not all views under the umbrella of scientifically valid can withstand experimental conditions. However, that is a topic for another time, another blog entry.

 

I would postulate that considering how the human mind seeks to find patterns to make sense of the environmental stimuli would explain our susceptibility to such a fallacy.  The same can be said for optical illusions and a plethora of other illusions and flaws in reasoning. It can be said that sometimes we lock on to details that may not necessarily be the focal point of causation. When faced with the amount of sensory input we are faced with it may be necessary to filter out details that are perceived as redundant or immaterial to our most pertinent needs. The need to quickly process information then make even quicker decisions enviably will lead to hasty and rash attributions. In some instances, it may be advantageous to make snap decisions in such a manner in others only leads to folly.  In activities that inherently solicit impulsivity such as gambling would make it easy to have this propensity for heuristic thinking to be more a hindrance than a help.