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I hear we’re putting the band back toether
"Il y a quelque chose de plus puissant que la force brutale des baïonnettes: c'est l'idée dont le temps est venu et l'heure est sonnée." Victor Hugo
PASCAL'S WAGER: RELIGION, MATH & WAR
Mathematician Blaise Pascal famously said that either there is a God or is not, and that the burdens of being a Christian are light, few, even none. Now if there is no god then belief or disbelief would not change that fact. But if there is the Christian God then the costs of faith being so low and the benefits being so great mean one ought believe.
While Pascal is right about the cost benefit analysis his view doesn't consider things like: what if some other God or Gods are the truth?
About 1/3 or even half of what is in the following video is just wrong. However, Rudyard is wrong in non catastrophic and very interesting ways. And some of it is right. Meanwhile, poor Rudyard Lynch fails to think about disinformation... a lot of so called "psychic" things are simply disinformation and many others are scams. Some people lie, a few lie It is much easier if your opponent is looking into the wrong things to understand your greater insights which lead you to win war after war.
To reiterate things I have already said:
The entire universe might well be a self aware living being, composed of other living beings, just as a human or animal is composed of living cells.
You yourself are god, or at least part of the god form.
Everything is alive, even "things". I'm an animist.
I also happen to believe in rebirth, the transmigration of souls, that this is one life in a series of many, many lives, which generally seek to grow to some greater level of awareness.
Controlled remote viewing is the one reproducible success of CIA investigations into paranormal/psychic things. We get much better predictions from large language models with specialized expert system modules, at least about macro phenomena such as war, than we do from various divination methods.
All States observed first arose out of the extended family, justifying State power by analogies to the family and religion. The questions of religions are ultimately questions of state, but it is also evident that agnosticism, secularism, or even atheism emerge as better tools for marshalling state power at the continental level. Secular tolerance emerged as the dominante view because they are more effective at organizing production and stifling conflict. Basically if Falun Gong, the Caliphate, Zion, and the Kingdom of God all came into power they would fairly quickly wind up in Crusade, Jihad, Genocide because religion is that powerful: it makes men willing to die and justifies every inhumane action. Religion makes war about eternal salvation or eternal damnation, and makes demonizing the enemy all too easy, and thus makes warfare extreme, total, and ineffective. Look at the Caliphate's tactics! They failed for fairly obvious reasons.
Sleep learning: you can reinforce and review already learned materials while sleeping. Example: I run through an hour of language rote sentence examples (vocabulary word, sample sentence, for an hour or so). Then, at night, I put that same rote on loop and listen to it allllllllll night. This does work. But just trying to learn new materials while sleeping does not.
Human animal communication is a reality, though dogs and cats have very limited vocabulary, just about 300 words or fewer. We will speak to alien civilizations before we meet them and most human models of communication are either limited, or of limited use.
In happier times people sometimes call the CIA "The Department of Dirty Tricks". Infiltrators and outright traitors in war time can be outright killed, though I think that extreme. It really would suck if someone started leaving dead birds at the traitors door. I would especially hate it if the traitor's tires of her car were slashed, or if someone started leaving dog shit in places like her dining table. What I mean is, there are ways short of outright killing or even beating someone down to get compliance.
Russia creates three special economic zones:
https://www.interfax.ru/russia/949581
Translation: Russian economy ruined, tries to buy off local thugs with rules changes that wouldn't get enforced anyway thanks to bribery, hopes to prevent secessionist tendencies, only reinforces them.
Sanctions work.
So does bombing "Russian" plane factories and ,burning "Russian" oil fields, and gas pipelines Not sorry.
China
Learn Chinese!
Math is war. Math is also business!
I want to compare four math concepts using chatgpt 3.5:
1) The gambler's fallacy
2) The law of probabilities
3) Regression to the mean
4) Bayes' theorem
because this comparison reveals squints, through which we can shoot our enemies dead or make money, your choice. How? Simple! This comparison will reveal
(1) paradoxes -- flaws in our own thinking, usually self-contradictions, and
(2) hiden relationships among these four concepts -- things we might not have seen, which by perceiving improve our ability to attack our enemy or defend ourselves. In business terms, trading opportunities or improvements in our production processes. These hidden relationships may also be unperceived by our opponent or competitor and thus may reveal weaknesses, points for attack, potential flaws in our "competitor's" thinking and
(3) enythemes, our implicit unstated presumptions in our own thinking, which we may or may not be correct about but might want to make explicit or be reminded of to avoid our own errors and determine whether our opponents has the same presumptions or not.
Plus I'm just a curious fucking monkey.
Hey! Is that a bomb? Let's compare notes! I am curious!
So... I asked chatgpt what it thinks, corrected/edited the results added some comments *thusly* and end the essay with a glaring chatgpt error, which I have left uncorrected and which you can correct if you like! Chatgpt 3.5 got these queries about 90% right, in other words wrong. Publicly available large language models without further local training can at least be a useful assistive technology but must be viewed critically. Let's get into it!
Regression to the mean compared with the gambler's fallacy:
Regression to the mean and the gambler's fallacy are two concepts that involve probabilities and expectations, but they work in different ways.
Regression to the mean:
Extreme outcomes are likely to be followed by more typical outcomes.
For example, if a basketball player scores way above their average in one game, they're likely to score closer to their average in the next game. Similarly, if they perform poorly in one game, they're likely to do better in the next.
It's about things balancing out over time, tending towards the average or mean.
Gambler's fallacy:
This is the mistaken belief that if something happens more frequently than the statistical average normal during a certain period, it will happen less frequently in the future, or vice versa.
For example, if a gambler flips a coin and it lands on heads five times in a row, they might believe that tails is "due" to come up next. But in reality, each coin flip is independent, so the outcome of one flip doesn't affect the outcome of the next.
It's a misunderstanding of probability, assuming that past outcomes influence future ones when they don't.
While regression to the mean suggests that extreme outcomes tend to balance out over time, the gambler's fallacy mistakenly assumes that past outcomes influence future ones, when they're actually independent events.
*This was a satisfactory response but didn't bring out the fact that regression to the mean is usually applied to non-stochastic processes whereas the gambler's fallacy misapprehends the nature of stochastic processes. The gambler's fallacy is a set of cognitive errors, most of which are supernatural: believing in "hot" and "cold" "streaks" thinking "lady luck" may be propitiated, that one's willpower and telekinesis influence the throw of the dice.*
CALCULATING THE PROBABILITIES OF TWO SIX SIDED DICE ROLLS:
To determine the likeliest average outcome of throwing two dice ten times, we first need to understand the possible outcomes when rolling two dice:
Each die has six sides, so there are
6
×
6
=
36
6×6=36 possible outcomes.
The sum of the two dice can range from 2 (when both dice show a 1) to 12 (when both dice show a 6).
Now, let's find the probabilities of each possible outcome:
The number of ways to get each sum is as follows:
Sum: 2 3 4 5 6 7 8 9 10 11 12
Ways: 1 2 3 4 5 6 5 4 3 2 1
The total number of outcomes is 36.
So, the probabilities of each sum are:
Sum: 2 3 4 5 6 7 8 9 10 11 12
Prob: 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36
The average sum when rolling two dice is indeed 7.
This is because each die has an equal probability of landing on any number from 1 to 6, and the sum of two dice is most likely to occur in the middle of the range of possible sums, which is 7.
So, if you were to roll two dice many times and calculate the average sum of those rolls, it would tend toward 7.
*ChatGPT3.5 seems to have gotten this one right*
The gamblers fallacy and probabilities:
The Gambler's Fallacy is an incorrect understanding of probability. It's the mistaken belief that past outcomes influence future ones when they're actually independent events. Each roll of the dice is an independent event, meaning the outcome of one roll doesn't influence the outcome of the next. So, if you rolled two dice and got a certain sum, the next roll still has the same probability distribution regardless of the previous outcome.
The average value of 7 for the sum of two six-sided dice comes from the fact that each die has an equal probability of landing on any number from 1 to 6. Since there are 36 possible outcomes when rolling two dice, each possible sum from 2 to 12 has a different probability of occurring. However, when you add up all these probabilities, the sum tends towards 7 as the most likely outcome. This is because there are more combinations of dice rolls that result in a sum of 7 than any other sum.
So, the average value of 7 is a result of the probabilities associated with the outcomes of rolling two dice, not because of any influence from past rolls or the Gambler's Fallacy.
*basically correct, still didn't bring out the differences between the gambler's fallacy and regression to the mean. Maybe my prompt sucked?*
THE LAW OF AVERAGES, THE GAMBLERS FALLACY, AND BAYES THEOREM
The Gambler's Fallacy, the Law of Averages, and Bayes' Theorem offer different perspectives on probability and decision-making. Despite their apparent differences, these concepts often intersect, shaping our understanding of randomness, risk, and inference. Examining these concepts through illustrative examples elucidates their relationships and implications.
The Gambler's Fallacy:
The Gambler's Fallacy is a cognitive bias that arises from the mistaken belief that past independent events influence future outcomes. It suggests that if a certain event has occurred frequently in the past, it is less likely to occur in the future, or vice versa. One classic example of the Gambler's Fallacy is in games of chance, such as roulette. A gambler may believe that after several consecutive red outcomes, black is "due" to come up next. However, each spin of the roulette wheel is an independent event, and the probability of landing on red or black remains constant regardless of past outcomes.
The Gambler's Fallacy arises from the mistaken belief that past independent events influence future outcomes. It suggests that if a certain event has occurred frequently in the past, it is less likely to occur in the future, or vice versa. One implication of the Gambler's Fallacy is that it can lead individuals to make irrational decisions in situations involving randomness. For example, in gambling scenarios like roulette, a player may incorrectly believe that after several consecutive red outcomes, black is "due" to come up next. However, each spin of the roulette wheel is an independent event, and the probability of landing on red or black remains constant regardless of past outcomes.
This erroneous inference can lead individuals to make poor decisions based on faulty reasoning, as they may incorrectly assume that previous outcomes will affect the likelihood of future events. The gambler's fallacy often reinforces other cognitive biases such as the escalation of commitment and the sunk costs' fallacy.
The Law of Averages:
Contrary to the Gambler's Fallacy, the Law of Averages states that over a large number of trials, the observed outcomes will tend to approach the expected value or average. This principle underpins much of statistical reasoning, suggesting that random fluctuations in outcomes will be balanced out over time. For instance, if you flip a fair coin thousands of times, the proportion of heads and tails will converge to 0.5 each, reflecting the true probability of each outcome.
The Law of Averages states that over a large number of trials, the observed outcomes will tend to approach the expected value or average. This principle suggests that random fluctuations in outcomes will be balanced out over time. One implication of the Law of Averages is that it provides a basis for understanding the long-term behavior of random processes. For example, if you flip a fair coin thousands of times, the proportion of heads and tails will converge to 0.5 each, reflecting the true probability of each outcome.
Erroneous Inference: Despite its name, the Law of Averages does not imply that outcomes will necessarily balance out in the short term or in every individual trial. This misconception can lead to erroneous expectations about the outcomes of random events, as individuals may mistakenly assume that deviations from the expected average in the short term are indicative of a trend that will eventually revert to the mean.
*A good way to think of it is like this: the likelist result of 2 dice thrown is 7. If you've had a series of 2s, which is really improbable, NOT impossible, you should still bet on 7. And if you get a string of 7s, the same thing.. *
Bayes' Theorem:
Bayes' Theorem provides a framework for updating beliefs or probabilities based on new evidence. It quantifies how our confidence in a hypothesis should change in light of new information. Formally, Bayes' Theorem states that the posterior probability of a hypothesis given observed data is proportional to the prior probability of the hypothesis multiplied by the likelihood of the data given the hypothesis. This theorem is invaluable in fields like machine learning, where it is used for probabilistic inference and decision-making.
Canonical Examples:
The Monty Hall game is a probabilistic puzzle named after the host of the game show "Let's Make a Deal." It goes like this:
A contestant chooses one of three doors, behind one of which is a car, and the other two have goats. The contestant doesn't know which door has the car, and the host, Monty Hall, knows where the car is. Monty then opens one of the two remaining doors, revealing a goat. The contestant must then decide whether to stick with their original choice or switch to the last unopened door. The question is, what is the best strategy for the contestant to maximize their chances of winning the car?
The twist in the game is that Monty knows where the car is, and he will always open a door with a goat behind it, giving the illusion that the odds of winning are now 50-50 between the two remaining doors.
The twist in the game is that Monty knows where the car is, and he will always open a door with a goat behind it, giving the illusion that the odds of winning are now 50-50 between the two remaining doors. However, the actual probability of the car being behind each door has not changed, and the strategic move for the contestant depends on whether they switch or stick with their original choice.
*Our intuition might suggest that there is a 50-50 chance of winning regardless of whether the contestant switches doors. This is because of an enthymematic assumption that the distribution of goats is random, as opposed to the contrary enthymemes, that Monty is trying to trick us or that Monty is trying to help us. The non-elucidation of these three enthymemes is why Monty Hall is a shitty way to explain Bayes, unless you also explain the enthymemes and the concept of Paradox (See Gödel, see also Willard O V Quine).*
Bayes' Theorem reveals that switching doors doubles the contestant's chances of winning. Initially, the prior probability of choosing the correct door is 1/3. When the host reveals a goat behind one of the other doors, the posterior probability that the unchosen door hides the prize increases to 2/3. This counterintuitive result demonstrates how Bayesian reasoning can lead to better decision-making.
The Monty Hall problem also illustrates the pitfalls of succumbing to the Gambler's Fallacy. Some contestants might believe that sticking with their initial choice is justified because they have already made their selection and switching now would not change the outcome. However, this line of reasoning fails to consider the new information provided by the host's action. Each door has an independent probability of concealing the prize, and this probability shifts based on the host's revelation.
Furthermore, the Law of Averages comes into play when considering the long-term outcomes of the Monty Hall problem. If the game were repeated many times, the Law of Averages predicts that contestants who consistently switch doors would win approximately twice as often as those who stubbornly stick with their initial choice. This convergence towards the expected value demonstrates how random fluctuations are smoothed out over multiple trials, reinforcing the validity of Bayesian reasoning over time.
Bayes' Theorem:
Bayes' Theorem provides a framework for updating beliefs or probabilities based on new evidence. It quantifies how our confidence in a hypothesis should change in light of new information. One implication of Bayes' Theorem is that it enables rational decision-making in uncertain or probabilistic environments. By systematically updating prior beliefs with new evidence, individuals can make more informed decisions and avoid biases that may arise from relying solely on intuition or past experiences.
Applying Bayes' Theorem to the Monty Hall Problem: Strategies for Winning in Three Scenarios"
The three scenarios are:
1. Monty wants the player to lose
2. Monty wants the player to win
3. Monty does not care whether the player wins or loses
By understanding the probabilities involved in each scenario and using Bayes' theorem, the player can make informed decisions that maximize their chances of winning.
Bayes' theorem is a fundamental concept in probability theory that provides a way to update probabilities based on new information. It can be applied to various problems, including the Monty Hall problem. Here are the three cases you mentioned and how Bayes' theorem applies to each of them:
1. Case One: Monty wants the player to lose
In this case, Monty knows which door has the car behind it and will always open a door with a goat behind it. The player's initial probability distribution for the location of the car is uniform (1/3) across all three doors. When Monty opens a door with a goat behind it, the player's probability distribution should update to reflect the fact that the car cannot be behind that door.
Using Bayes' theorem, we can calculate the updated probability distribution as follows:
P(car behind door A | Monty opened door B) = P(Monty opened door B | car behind door A) \* P(car behind door A) / P(Monty opened door B)
where:
- P(car behind door A) is the prior probability of the car being behind door A, which is 1/3.
- P(Monty opened door B | car behind door A) is the conditional probability of Monty opening door B given that the car is behind door A. Since Monty wants the player to lose, he will always open a door with a goat behind it, so this probability is 1.
- P(Monty opened door B) is the normalizing constant or evidence term, which is equal to the probability of Monty opening door B regardless of where the car is located. This probability is also 1/3 since there are three possible locations for the car and Monty has no preference among them.
Therefore, the updated probability distribution after Monty opens a door with a goat behind it is:
P(car behind door A | Monty opened door B) = 0
P(car behind door B | Monty opened door B) = 1/2
P(car behind door C | Monty opened door B) = 1/2
The player should now choose door C, as it has the highest probability of having the car behind it.
2. Case Two: Monty wants the player to win
In this case, Monty knows which door has the car behind it and will always open a door with a goat behind it, but only if the car is not behind that door. If the car is behind the door Monty opens, he will pretend that it's a goat. The player's initial probability distribution is still uniform (1/3) across all three doors.
When Monty opens a door with a goat behind it, the player's probability distribution should update to reflect the fact that the car cannot be behind that door. However, since Monty wants the player to win, the probability of the car being behind the other two doors increases.
Using Bayes' theorem again:
P(car behind door A | Monty opened door B) = P(Monty opened door B | car behind door A) \* P(car behind door A) / P(Monty opened door B)
where:
- P(car behind door A) is the prior probability of the car being behind door A, which is 1/3.
- P(Monty opened door B | car behind door A) is the conditional probability of Monty opening door B given that the car is behind door A. Since Monty wants the player to win, this probability is 0, as Monty would never open a door with the car behind it.
- P(Monty opened door B) is the normalizing constant or evidence term, which is equal to the probability of Monty opening door B regardless of where the car is located. This probability is 1/3 since there are three possible locations for the car and Monty has no preference among them.
Therefore, the updated probability distribution after Monty opens a door with a goat behind it is:
P(car behind door A | Monty opened door B) = 1/2
P(car behind door B | Monty opened door B) = 0
P(car behind door C | Monty opened door B) = 1/2
Now, the player should switch their choice to door A, as it has the highest probability of having the car behind it.
3. Case Three: Monty does not care whether the player wins or loses
In this case, Monty randomly selects a door to open, without any knowledge about the location of the car. The player's initial probability distribution is still uniform (1/3)
In the third case where Monty does not care whether the player wins or loses, Bayes' theorem does not provide any advantage in guessing where the car is located.
When Monty is indifferent, the probability of him opening any particular door is equal (1/3), and the probability of the car being behind any particular door is also equal (1/3). Therefore, the player's initial probability distribution is already accurate, and there is no new information to update the probabilities.
In this scenario, the player's strategy should be to stick with their original choice, as switching doors does not provide any advantage. This is because Monty's actions do not provide any additional information that can help the player make a better decision.
So, while Bayes' theorem is useful in cases where Monty wants the player to win or lose, it does not help in situations where Monty is indifferent.
WHAT CAUSES THE GAMBLERS FALLACY AND WHY IS IT SO COMMON?
The Gambler's Fallacy is a cognitive bias that arises from the human tendency to perceive patterns and impose order on random events. It leads individuals to believe that past independent events influence future outcomes, contrary to the principles of probability theory. This fallacy, rooted in our innate cognitive processes, can be understood through the lens of personification and pareidolia – phenomena in which human beings attribute human-like characteristics or patterns to non-human entities or random stimuli. By exploring examples of personification in religious beliefs, pareidolia in visual perception, and the Gambler's Fallacy in decision-making, we can gain insight into the mechanisms underlying these cognitive biases and their implications for human behavior.
Personification in Religious Beliefs:
Religious beliefs often involve personifying abstract concepts or natural phenomena, attributing human-like qualities or intentions to deities, spirits, or forces of nature. For example, ancient cultures personified natural elements such as the sun, moon, and rain as gods or goddesses with distinct personalities and motivations. Similarly, monotheistic religions often depict a single deity with anthropomorphic traits such as wisdom, compassion, and judgment. These personifications serve to make the abstract concepts more relatable and understandable to human beings, allowing individuals to ascribe agency and meaning to natural phenomena beyond their control.
Pareidolia in Visual Perception:
Pareidolia refers to the tendency to perceive meaningful patterns or faces in random stimuli, such as clouds, rock formations, or inanimate objects. This phenomenon arises from the brain's inherent predisposition to recognize familiar shapes and structures, even when they are not actually present. For example, individuals may perceive a face in the arrangement of shadows and light on a wall or interpret random noise as voices or messages in audio recordings. Pareidolia highlights the brain's remarkable ability to detect patterns and make sense of ambiguous stimuli, but it can also lead to misinterpretations and false perceptions when applied to random or meaningless stimuli.
The Gambler's Fallacy in Decision-Making:
The Gambler's Fallacy, a cognitive bias related to probability theory, reflects the human tendency to perceive patterns and predictability in random events. It arises from the mistaken belief that past outcomes influence future probabilities, leading individuals to expect deviations from randomness and anticipate "corrections" to balance out perceived patterns. For example, in gambling scenarios such as roulette or coin tosses, individuals may believe that after a series of consecutive outcomes of the same type, the next outcome is more likely to be the opposite, leading to irrational betting behavior. The Gambler's Fallacy illustrates how our cognitive biases can influence decision-making processes, leading to erroneous beliefs and behaviors based on faulty reasoning.
The Gambler's Fallacy, rooted in the human tendency to perceive patterns and impose order on random events, can be understood through the concepts of personification and pareidolia. By attributing human-like qualities to non-human entities or detecting meaningful patterns in random stimuli, individuals may fall prey to cognitive biases that lead to erroneous beliefs and behaviors. Understanding the underlying mechanisms of these biases can help us recognize and mitigate their influence on decision-making processes, promoting more rational and informed choices in various domains of life.
WHY BAYES THEOREM WORKS
In cases where human will plays a part, such as decision-making or subjective judgments, Bayes' Theorem can still be used to update probabilities based on new information or evidence. However, there are several important considerations to keep in mind:
1. **Subjectivity and Prior Beliefs:** In Bayesian inference, prior beliefs or subjective probabilities are an essential component of the analysis. When human will or subjective judgments are involved, individuals may have prior beliefs or biases that influence their decision-making process. These prior beliefs can be incorporated into the analysis as the initial probability distribution before observing any new evidence. The strength of prior beliefs and their impact on the posterior probability distribution will depend on the context and the individual's level of confidence in their beliefs.
2. **Updating Beliefs with Evidence:** Bayes' Theorem allows for the systematic updating of beliefs based on new evidence or observations. In cases where human will plays a part, new evidence may come in the form of actions, decisions, or observed behaviors. By incorporating this new evidence into the analysis, individuals can update their beliefs and make more informed decisions. However, it's important to recognize that the interpretation of evidence and its relevance to the problem at hand may be subjective and influenced by individual perspectives or biases.
3. **Complexity and Uncertainty:** In real-world scenarios involving human will, decision-making processes can be complex and uncertain. Bayes' Theorem provides a formal framework for reasoning under uncertainty and can help individuals make decisions based on available evidence and probabilistic reasoning. However, the complexity of human behavior and the presence of unknown or unobservable factors can make it challenging to accurately model and predict outcomes. Bayesian analysis can provide valuable insights into the decision-making process but may also require careful consideration of model assumptions and limitations.
THE GAMBLERS FALLACY AND BAYES THEOREM ARE MIRRORS!
*We saw the gamblers fallacy basically arises out of personification of events, thinking there is "Lady Luck" to propitiate, that there are "hot" and "cold" "streaks", ignoring the fact that the laws of physics are what determine dice rolls, not probabilities. Physically speaking of throwing a die will always have a 1/6 chance of falling on six, no matter how many or few sixes preceded this throw.
At the same time, this appears paradoxal because of "the law of probabilities" and "regression to the mean". In the former case, probabilities of truly stochastic processes are only likelihoods, not certainties. In the latter case, regression to the mean arises out of some human process as opposed to merely stochastic ones. If I tend to get 90% scores or better on a test and one day do really badly, say because I am sick, it is no surprise that I will regress to my usual good performance once better. Likewise, the "new" information impacting unknown possible future information may also depend on human will.*
1. **Law of Probabilities and Stochastic Processes:**
The law of probabilities, also known as the law of large numbers, states that the observed outcomes of truly stochastic processes tend to converge towards their expected probabilities as the number of trials increases. This means that while individual outcomes may vary in the short term, the long-term average will approach the expected value dictated by the underlying probability distribution. For example, in the context of coin flips, the probability of getting heads is 0.5. While a short sequence of coin flips may result in an imbalance of heads and tails, over a large number of flips, the proportion of heads will tend towards 0.5.
2. **Regression to the Mean:**
Regression to the mean refers to the tendency for extreme outcomes to move closer to the average over time. This phenomenon arises in situations where extreme outcomes are often due to chance or random variability rather than systematic factors. For instance, if a student consistently performs exceptionally well on tests but performs poorly on one particular test due to illness or other temporary factors, it's likely that their performance will revert back to their usual level of performance once the temporary factors are resolved. This is because extreme outcomes are often outliers and not representative of the individual's true ability.
3. **Bayes' Theorem and Human Will:**
Bayes' Theorem provides a framework for updating beliefs or probabilities based on new evidence. In cases where human will plays a part, such as decision-making or subjective judgments, Bayes' Theorem can still be applied to update probabilities based on new information or observations. However, the interpretation of evidence and its relevance to the problem at hand may be subjective and influenced by individual perspectives or biases. For example, if a person flips a coin 100 times and gets 100 heads in a row, their belief in the fairness of the coin may change based on this new evidence. Bayes' Theorem allows for the systematic updating of beliefs based on new evidence, even in situations where human will is involved.
2. **Regression to the Mean:**
Regression to the mean is a statistical phenomenon where extreme observations tend to move closer to the average upon repeated measurements. It occurs because extreme values are often due to random variability rather than systematic factors. For example, if a student scores exceptionally high on a test due to luck or random chance, it's likely that their subsequent scores will regress towards the mean (the average score) over time.
3. **Law of Averages:**
The law of averages, also known as the law of large numbers, states that the observed outcomes of random processes tend to converge towards the expected value as the number of trials increases. It describes the tendency for the average outcome to approach the theoretical or expected value over a large number of trials. For example, in the context of coin flips, the law of averages predicts that the proportion of heads will approach 0.5 as the number of flips increases, reflecting the true probability of heads.
4. **Gambler's Fallacy:**
The Gambler's Fallacy is a cognitive bias that arises from the mistaken belief that past independent events influence future outcomes. It suggests that if a certain event has occurred frequently in the past, it is less likely to occur in the future, or vice versa. For example, in gambling scenarios such as roulette, a player may incorrectly believe that after several consecutive red outcomes, black is "due" to come up next. However, each spin of the roulette wheel is an independent event, and the probability of landing on red or black remains constant regardless of past outcomes.
Now, let's address any apparent contradictions:
- **Bayes' Theorem vs. Gambler's Fallacy:** Bayes' Theorem and the Gambler's Fallacy may appear contradictory because they describe different aspects of probabilistic reasoning. Bayes' Theorem provides a rational framework for updating beliefs based on new evidence, taking into account prior probabilities and the likelihood of the data. In contrast, the Gambler's Fallacy is a cognitive bias that leads individuals to erroneously believe that past outcomes influence future probabilities. The contradiction arises because Bayes' Theorem acknowledges the independence of events, whereas the Gambler's Fallacy violates this principle by assuming a relationship between past and future outcomes.
- **Regression to the Mean vs. Gambler's Fallacy:** Regression to the mean and the Gambler's Fallacy may also seem contradictory because they describe different patterns of behavior in response to random variability. Regression to the mean recognizes that extreme outcomes are often due to chance and tend to revert towards the average over time. In contrast, the Gambler's Fallacy suggests that past outcomes influence future probabilities, leading individuals to expect deviations from randomness. The contradiction arises because regression to the mean is based on the statistical properties of random processes, whereas the Gambler's Fallacy is a cognitive bias that disregards these properties.
While these concepts may appear contradictory on the surface, they are not inherently incompatible. Instead, they address different aspects of probabilistic reasoning and decision-making. Bayes' Theorem provides a rational framework for updating beliefs based on new evidence, regression to the mean describes statistical tendencies observed in random processes, the law of averages describes the convergence of observed outcomes towards expected values, and the Gambler's Fallacy highlights cognitive biases that can lead to erroneous beliefs and behaviors. By understanding the nuances of each concept, we can navigate probabilistic scenarios more effectively and make more informed decisions.
The Maths of Bayes' Theorem: CHAT GPT FUCKS IT UP AGAIN!
Bayes' Theorem can be used to update our belief in the probability of the coin landing on heads given the new evidence of observing three consecutive heads. Let's denote:
- H: The event that the coin lands on heads.
- E: The evidence of observing three consecutive heads.
We want to calculate P(H|E), the posterior probability of the coin landing on heads given the evidence E.
Bayes' Theorem states:
P(H|E) = P(E|H) * P(H)/P(E)
where:
- P(H) is the prior probability of the coin landing on heads (before observing any evidence).
- P(E|H) is the likelihood of observing the evidence given that the coin lands on heads.
- P(E) is the total probability of observing the evidence, considering all possible hypotheses (landing on heads or tails).
Let's break down each component:
1. P(H): The prior probability of the coin landing on heads. If we assume the coin is fair and unbiased, P(H) = 0.5
2. P(E|H): The likelihood of observing three consecutive heads given that the coin lands on heads. Since each coin toss is independent, the probability of getting heads three times in a row is (0.5)^3 = 0.125
3. P(E): The total probability of observing three consecutive heads, considering all possible hypotheses (landing on heads or tails). This can be calculated by considering both cases:
P(E) = P(E|H) * P(H) + P(E| ¬H) * P(¬H)
P(E) = (0.125) * (0.5) + (0) * (0.5) (since getting three consecutive tails is impossible)
- P(E) = 0.0625+
Now, we can calculate the posterior probability:
P(H|E) = (0.125) * (0.5) / 0.0625
= 0.0625/0.0625
= 1
So, according to Bayes' Theorem, after observing three consecutive heads, the posterior probability of the coin landing on heads is 1 or 100%. This means that, given the evidence, we are certain that the coin is biased towards landing on heads.
SPOT THE FLAWS WIN THE PRIZE.
Conclusion:
In conclusion, the Gambler's Fallacy, the Law of Averages, and Bayes' Theorem offer distinct yet interconnected perspectives on probability and decision-making. Through canonical examples such as the Monty Hall problem, we have seen how these concepts influence each other and shape our understanding of randomness, risk, and inference. While the Gambler's Fallacy highlights the dangers of relying on past events to predict future outcomes, the Law of Averages reminds us of the stabilizing effect of large sample sizes. Bayes' Theorem provides a principled framework for updating beliefs in the face of new evidence, offering a path towards more rational decision-making. By embracing these concepts and their interrelationships, we can navigate uncertainty with greater clarity and confidence in both theoretical and practical settings.
*In all one can clearly use chatgpt3.5 as a writing aid, but one must be careful. Chatgpt tells you what that large language model BELIEVES to be the truth, not what is in fact certain truth.