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In an attempt to clarify her answer, she proposed a shell game to illustrate: "You look away, and I put a pea under one of three shells. Then I ask you to put your finger on a shell. The odds that your choice contains a pea are , agreed? Then I simply lift up an empty shell from the remaining other two. As I can (and w...
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Vos Savant commented that, though some confusion was caused by "some" readers' not realizing they were supposed to assume that the host must always reveal a goat, almost all her numerous correspondents had correctly understood the problem assumptions, and were still initially convinced that vos Savant's answer ("switch...
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When first presented with the Monty Hall problem, an overwhelming majority of people assume that each door has an equal probability and conclude that switching does not matter. Out of 228 subjects in one study, only 13% chose to switch. In his book "The Power of Logical Thinking", cognitive psychologist writes: "No oth...
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Most statements of the problem, notably the one in "Parade", do not match the rules of the actual game show and do not fully specify the host's behavior or that the car's location is randomly selected. However, Krauss and Wang argue that people make the standard assumptions even if they are not explicitly stated.
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Although these issues are mathematically significant, even when controlling for these factors, nearly all people still think each of the two unopened doors has an equal probability and conclude that switching does not matter. This "equal probability" assumption is a deeply rooted intuition. People strongly tend to thin...
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The problem continues to attract the attention of cognitive psychologists. The typical behavior of the majority, i.e., not switching, may be explained by phenomena known in the psychological literature as:
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Experimental evidence confirms that these are plausible explanations that do not depend on probability intuition. Another possibility is that people's intuition simply does not deal with the textbook version of the problem, but with a real game show setting. There, the possibility exists that the show master plays dece...
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As already remarked, most sources in the field of probability, including many introductory probability textbooks, solve the problem by showing the conditional probabilities that the car is behind door 1 and door 2 are and (not and ) given that the contestant initially picks door 1 and the host opens door 3; various way...
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Among these sources are several that explicitly criticize the popularly presented "simple" solutions, saying these solutions are "correct but ... shaky", or do not "address the problem posed", or are "incomplete", or are "unconvincing and misleading", or are (most bluntly) "false".
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Sasha Volokh (2015) wrote that "any explanation that says something like 'the probability of door 1 was , and nothing can change that...' is automatically fishy: probabilities are expressions of our ignorance about the world, and new information can change the extent of our ignorance."
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Some say that these solutions answer a slightly different question – one phrasing is "you have to announce "before a door has been opened" whether you plan to switch".
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The simple solutions show in various ways that a contestant who is determined to switch will win the car with probability , and hence that switching is the winning strategy, if the player has to choose in advance between "always switching", and "always staying". However, the probability of winning by "always" switching...
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The answer to the first question is , as is correctly shown by the "simple" solutions. But the answer to the second question is now different: the conditional probability the car is behind door 1 or door 2 given the host has opened door 3 (the door on the right) is . This is because Monty's preference for rightmost doo...
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In Morgan "et al.", four university professors published an article in "The American Statistician" claiming that vos Savant gave the correct advice but the wrong argument. They believed the question asked for the chance of the car behind door 2 "given" the player's initial pick for door 1 and the opened door 3, and the...
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In an invited comment and in subsequent letters to the editor, Morgan "et al" were supported by some writers, criticized by others; in each case a response by Morgan "et al" is published alongside the letter or comment in "The American Statistician". In particular, vos Savant defended herself vigorously. Morgan "et al"...
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There is disagreement in the literature regarding whether vos Savant's formulation of the problem, as presented in "Parade", is asking the first or second question, and whether this difference is significant. Behrends concludes that "One must consider the matter with care to see that both analyses are correct"; which i...
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Among the simple solutions, the "combined doors solution" comes closest to a conditional solution, as we saw in the discussion of approaches using the concept of odds and Bayes' theorem. It is based on the deeply rooted intuition that "revealing information that is already known does not affect probabilities". But, kno...
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The simple solutions above show that a player with a strategy of switching wins the car with overall probability , i.e., without taking account of which door was opened by the host. In accordance with this, most sources in the field of probability calculate the conditional probabilities that the car is behind door 1 an...
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If we assume that the host opens a door at random, when given a choice, then which door the host opens gives us no information at all as to whether or not the car is behind door 1. In the simple solutions, we have already observed that the probability that the car is behind door 1, the door initially chosen by the play...
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By definition, the conditional probability of winning by switching given the contestant initially picks door 1 and the host opens door 3 is the probability for the event "car is behind door 2 and host opens door 3" divided by the probability for "host opens door 3". These probabilities can be determined referring to th...
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The conditional probability table below shows how 300 cases, in all of which the player initially chooses door 1, would be split up, on average, according to the location of the car and the choice of door to open by the host.
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Many probability text books and articles in the field of probability theory derive the conditional probability solution through a formal application of Bayes' theorem; among them books by Gill and Henze. Use of the odds form of Bayes' theorem, often called Bayes' rule, makes such a derivation more transparent.
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Initially, the car is equally likely to be behind any of the three doors: the odds on door 1, door 2, and door 3 are . This remains the case after the player has chosen door 1, by independence. According to Bayes' rule, the posterior odds on the location of the car, given that the host opens door 3, are equal to the pr...
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Richard Gill analyzes the likelihood for the host to open door 3 as follows. Given that the car is "not" behind door 1, it is equally likely that it is behind door 2 or 3. Therefore, the chance that the host opens door 3 is 50%. Given that the car "is" behind door 1, the chance that the host opens door 3 is also 50%, b...
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In words, the information "which" door is opened by the host (door 2 or door 3?) reveals no information at all about whether or not the car is behind door 1, and this is precisely what is alleged to be intuitively obvious by supporters of simple solutions, or using the idioms of mathematical proofs, "obviously true, by...
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Consider the event "Ci", indicating that the car is behind door number "i", takes value "Xi", for the choosing of the player, and value "Hi", for the host opening the door. The player initially chooses door i = 1, C = X1 and the host opens door i = 3, C = H3.
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"P(H3|X1) = 1/2" because this expression only depends on "X1", not on any "Ci". So, in this particular expression, the choosing of the host does not depend on where the car is, and there's only two remaining doors once "X1" is chosen (for instance, "P(H1|X1) = 0"); and "P(Ci,Xi) = P(Ci)P(Xi)" because "Ci" and "Xi" are ...
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Then, if the player initially selects door 1, and the host opens door 3, we prove that the conditional probability of winning by switching is:
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From the Bayes' rule, we know that "P(A,B) = P(A|B)P(B) = P(B|A)P(A)". Extending this logic to multiple events, for example "A", "B" and "C", we get that we can play with the different subsets of "{A, B, C}" to calculate the probability of the intersection, as a tool to simplify the calculation of our conditional proba...
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Going back to Nalebuff, the Monty Hall problem is also much studied in the literature on game theory and decision theory, and also some popular solutions correspond to this point of view. Vos Savant asks for a decision, not a chance. And the chance aspects of how the car is hidden and how an unchosen door is opened are...
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Following Gill, a "strategy" of contestant involves two actions: the initial choice of a door and the decision to switch (or to stick) which may depend on both the door initially chosen and the door to which the host offers switching. For instance, one contestant's strategy is "choose door 1, then switch to door 2 when...
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Elementary comparison of contestant's strategies shows that, for every strategy A, there is another strategy B "pick a door then switch no matter what happens" that dominates it. No matter how the car is hidden and no matter which rule the host uses when he has a choice between two goats, if A wins the car then B also ...
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Similarly, strategy A "pick door 1 then switch to door 2 (if offered), but do not switch to door 3 (if offered)" is dominated by strategy B "pick door 2 then always switch". A wins when door 1 conceals the car and Monty chooses to open door 2 or if door 3 conceals the car. Strategy B wins when either door 1 or door 3 c...
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Dominance is a strong reason to seek for a solution among always-switching strategies, under fairly general assumptions on the environment in which the contestant is making decisions. In particular, if the car is hidden by means of some randomization device – like tossing symmetric or asymmetric three-sided die – the d...
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Strategic dominance links the Monty Hall problem to game theory. In the zero-sum game setting of Gill, discarding the non-switching strategies reduces the game to the following simple variant: the host (or the TV-team) decides on the door to hide the car, and the contestant chooses two doors (i.e., the two doors remain...
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A simple way to demonstrate that a switching strategy really does win two out of three times with the standard assumptions is to simulate the game with playing cards. Three cards from an ordinary deck are used to represent the three doors; one 'special' card represents the door with the car and two other cards represen...
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The simulation can be repeated several times to simulate multiple rounds of the game. The player picks one of the three cards, then, looking at the remaining two cards the 'host' discards a goat card. If the card remaining in the host's hand is the car card, this is recorded as a switching win; if the host is holding a...
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Repeated plays also make it clearer why switching is the better strategy. After the player picks his card, it is "already determined" whether switching will win the round for the player. If this is not convincing, the simulation can be done with the entire deck. In this variant, the car card goes to the host 51 times o...
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A common variant of the problem, assumed by several academic authors as the canonical problem, does not make the simplifying assumption that the host must uniformly choose the door to open, but instead that he uses some other strategy. The confusion as to which formalization is authoritative has led to considerable acr...
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The version of the Monty Hall problem published in "Parade" in 1990 did not specifically state that the host would always open another door, or always offer a choice to switch, or even never open the door revealing the car. However, vos Savant made it clear in her second follow-up column that the intended host's behavi...
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Determining the player's best strategy within a given set of other rules the host must follow is the type of problem studied in game theory. For example, if the host is not required to make the offer to switch the player may suspect the host is malicious and makes the offers more often if the player has initially selec...
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Morgan "et al" and Gillman both show a more general solution where the car is (uniformly) randomly placed but the host is not constrained to pick uniformly randomly if the player has initially selected the car, which is how they both interpret the statement of the problem in "Parade" despite the author's disclaimers. B...
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D. L. Ferguson (1975 in a letter to Selvin) suggests an "N"-door generalization of the original problem in which the host opens "p" losing doors and then offers the player the opportunity to switch; in this variant switching wins with probability formula_5. This probability is always greater than formula_6, therefore s...
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Even if the host opens only a single door (formula_7), the player is better off switching in every case. As "N" grows larger, the advantage decreases and approaches zero.
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At the other extreme, if the host opens all losing doors but one ("p" = "N" − 2) the advantage increases as "N" grows large (the probability of winning by switching is , which approaches 1 as "N" grows very large).
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A quantum version of the paradox illustrates some points about the relation between classical or non-quantum information and quantum information, as encoded in the states of quantum mechanical systems. The formulation is loosely based on quantum game theory. The three doors are replaced by a quantum system allowing thr...
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The earliest of several probability puzzles related to the Monty Hall problem is Bertrand's box paradox, posed by Joseph Bertrand in 1889 in his "Calcul des probabilités". In this puzzle, there are three boxes: a box containing two gold coins, a box with two silver coins, and a box with one of each. After choosing a bo...
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The Three Prisoners problem, published in Martin Gardner's "Mathematical Games" column in "Scientific American" in 1959 is equivalent to the Monty Hall problem. This problem involves three condemned prisoners, a random one of whom has been secretly chosen to be pardoned. One of the prisoners begs the warden to tell him...
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Steve Selvin posed the Monty Hall problem in a pair of letters to "The American Statistician" in 1975. The first letter presented the problem in a version close to its presentation in "Parade" 15 years later. The second appears to be the first use of the term "Monty Hall problem". The problem is actually an extrapolati...
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A version of the problem very similar to the one that appeared three years later in "Parade" was published in 1987 in the Puzzles section of "The Journal of Economic Perspectives". Nalebuff, as later writers in mathematical economics, sees the problem as a simple and amusing exercise in game theory.
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"The Monty Hall Trap", Phillip Martin's 1989 article in "Bridge Today", presented Selvin's problem as an example of what Martin calls the probability trap of treating non-random information as if it were random, and relates this to concepts in the game of bridge.
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A restated version of Selvin's problem appeared in Marilyn vos Savant's "Ask Marilyn" question-and-answer column of "Parade" in September 1990. Though vos Savant gave the correct answer that switching would win two-thirds of the time, she estimates the magazine received 10,000 letters including close to 1,000 signed by...
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In November 1990, an equally contentious discussion of vos Savant's article took place in Cecil Adams's column "The Straight Dope". Adams initially answered, incorrectly, that the chances for the two remaining doors must each be one in two. After a reader wrote in to correct the mathematics of Adams's analysis, Adams a...
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The "Parade" column and its response received considerable attention in the press, including a front-page story in "The New York Times" in which Monty Hall himself was interviewed. Hall understood the problem, giving the reporter a demonstration with car keys and explaining how actual game play on "Let's Make a Deal" d...
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Dopamine (DA, a contraction of 3,4-dihydroxyphenethylamine) is a neuromodulatory molecule that plays several important roles in cells. It is an organic chemical of the catecholamine and phenethylamine families. Dopamine constitutes about 80% of the catecholamine content in the brain. It is an amine synthesized by remov...
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In popular culture and media, dopamine is often portrayed as the main chemical of pleasure, but the current opinion in pharmacology is that dopamine instead confers motivational salience; in other words, dopamine signals the perceived motivational prominence (i.e., the desirability or aversiveness) of an outcome, which...
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Outside the central nervous system, dopamine functions primarily as a local paracrine messenger. In blood vessels, it inhibits norepinephrine release and acts as a vasodilator (at normal concentrations); in the kidneys, it increases sodium excretion and urine output; in the pancreas, it reduces insulin production; in t...
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Several important diseases of the nervous system are associated with dysfunctions of the dopamine system, and some of the key medications used to treat them work by altering the effects of dopamine. Parkinson's disease, a degenerative condition causing tremor and motor impairment, is caused by a loss of dopamine-secret...
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A dopamine molecule consists of a catechol structure (a benzene ring with two hydroxyl side groups) with one amine group attached via an ethyl chain. As such, dopamine is the simplest possible catecholamine, a family that also includes the neurotransmitters norepinephrine and epinephrine. The presence of a benzene ring...
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Like most amines, dopamine is an organic base. As a base, it is generally protonated in acidic environments (in an acid-base reaction). The protonated form is highly water-soluble and relatively stable, but can become oxidized if exposed to oxygen or other oxidants. In basic environments, dopamine is not protonated. In...
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Dopamine is synthesized in a restricted set of cell types, mainly neurons and cells in the medulla of the adrenal glands. The primary and minor metabolic pathways respectively are:
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The direct precursor of dopamine, -DOPA, can be synthesized indirectly from the essential amino acid phenylalanine or directly from the non-essential amino acid tyrosine. These amino acids are found in nearly every protein and so are readily available in food, with tyrosine being the most common. Although dopamine is a...
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-Phenylalanine is converted into -tyrosine by the enzyme phenylalanine hydroxylase, with molecular oxygen (O) and tetrahydrobiopterin as cofactors. -Tyrosine is converted into -DOPA by the enzyme tyrosine hydroxylase, with tetrahydrobiopterin, O, and iron (Fe) as cofactors. -DOPA is converted into dopamine by the enzym...
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Dopamine itself is used as precursor in the synthesis of the neurotransmitters norepinephrine and epinephrine. Dopamine is converted into norepinephrine by the enzyme dopamine β-hydroxylase, with O and -ascorbic acid as cofactors. Norepinephrine is converted into epinephrine by the enzyme phenylethanolamine "N"-methylt...
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Some of the cofactors also require their own synthesis. Deficiency in any required amino acid or cofactor can impair the synthesis of dopamine, norepinephrine, and epinephrine.
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Dopamine is broken down into inactive metabolites by a set of enzymes—monoamine oxidase (MAO), catechol-"O"-methyl transferase (COMT), and aldehyde dehydrogenase (ALDH), acting in sequence. Both isoforms of monoamine oxidase, MAO-A and MAO-B, effectively metabolize dopamine. Different breakdown pathways exist but the m...
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In clinical research on schizophrenia, measurements of homovanillic acid in plasma have been used to estimate levels of dopamine activity in the brain. A difficulty in this approach however, is separating the high level of plasma homovanillic acid contributed by the metabolism of norepinephrine.
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Although dopamine is normally broken down by an oxidoreductase enzyme, it is also susceptible to oxidation by direct reaction with oxygen, yielding quinones plus various free radicals as products. The rate of oxidation can be increased by the presence of ferric iron or other factors. Quinones and free radicals produced...
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Dopamine exerts its effects by binding to and activating cell surface receptors. In humans, dopamine has a high binding affinity at dopamine receptors and human trace amine-associated receptor 1 (hTAAR1). In mammals, five subtypes of dopamine receptors have been identified, labeled from D1 to D5. All of them function a...
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Inside the brain, dopamine functions as a neurotransmitter and neuromodulator, and is controlled by a set of mechanisms common to all monoamine neurotransmitters. After synthesis, dopamine is transported from the cytosol into synaptic vesicles by a solute carrier—a vesicular monoamine transporter, VMAT2. Dopamine is st...
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Once in the synapse, dopamine binds to and activates dopamine receptors. These can be postsynaptic dopamine receptors, which are located on dendrites (the postsynaptic neuron), or presynaptic autoreceptors (e.g., the Dsh and presynaptic D receptors), which are located on the membrane of an axon terminal (the presynapti...
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In the brain the level of extracellular dopamine is modulated by two mechanisms: phasic and tonic transmission. Phasic dopamine release, like most neurotransmitter release in the nervous system, is driven directly by action potentials in the dopamine-containing cells. Tonic dopamine transmission occurs when small amoun...
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Inside the brain, dopamine plays important roles in executive functions, motor control, motivation, arousal, reinforcement, and reward, as well as lower-level functions including lactation, sexual gratification, and nausea. The dopaminergic cell groups and pathways make up the dopamine system which is neuromodulatory.
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Dopaminergic neurons (dopamine-producing nerve cells) are comparatively few in number—a total of around 400,000 in the human brain—and their cell bodies are confined in groups to a few relatively small brain areas. However their axons project to many other brain areas, and they exert powerful effects on their targets. ...
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The substantia nigra is a small midbrain area that forms a component of the basal ganglia. This has two parts—an input area called the pars compacta and an output area the pars reticulata. The dopaminergic neurons are found mainly in the pars compacta (cell group A8) and nearby (group A9). In humans, the projection of ...
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The ventral tegmental area (VTA) is another midbrain area. The most prominent group of VTA dopaminergic neurons projects to the prefrontal cortex via the mesocortical pathway and another smaller group projects to the nucleus accumbens via the mesolimbic pathway. Together, these two pathways are collectively termed the ...
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The posterior hypothalamus has dopamine neurons that project to the spinal cord, but their function is not well established. There is some evidence that pathology in this area plays a role in restless legs syndrome, a condition in which people have difficulty sleeping due to an overwhelming compulsion to constantly mov...
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The arcuate nucleus and the periventricular nucleus of the hypothalamus have dopamine neurons that form an important projection—the "tuberoinfundibular pathway" which goes to the pituitary gland, where it influences the secretion of the hormone prolactin. Dopamine is the primary neuroendocrine inhibitor of the secretio...
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The zona incerta, grouped between the arcuate and periventricular nuclei, projects to several areas of the hypothalamus, and participates in the control of gonadotropin-releasing hormone, which is necessary to activate the development of the male and female reproductive systems, following puberty.
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An additional group of dopamine-secreting neurons is found in the retina of the eye. These neurons are amacrine cells, meaning that they have no axons. They release dopamine into the extracellular medium, and are specifically active during daylight hours, becoming silent at night. This retinal dopamine acts to enhance ...
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The largest and most important sources of dopamine in the vertebrate brain are the substantia nigra and ventral tegmental area. These structures are closely related to each other and functionally similar in many respects. Both are components of the mid brain. The largest component of the basal ganglia is the striatum. ...
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Progress in understanding the functions of the basal ganglia has been slow. The most popular hypotheses, broadly stated, propose that the basal ganglia play a central role in action selection. The action selection theory in its simplest form proposes that when a person or animal is in a situation where several behavior...
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The basal ganglia can be divided into several sectors, and each is involved in controlling particular types of actions. The ventral sector of the basal ganglia (containing the ventral striatum and ventral tegmental area) operates at the highest level of the hierarchy, selecting actions at the whole-organism level. The ...
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Dopamine contributes to the action selection process in at least two important ways. First, it sets the "threshold" for initiating actions. The higher the level of dopamine activity, the lower the impetus required to evoke a given behavior. As a consequence, high levels of dopamine lead to high levels of motor activity...
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The second important effect of dopamine is as a "teaching" signal. When an action is followed by an increase in dopamine activity, the basal ganglia circuit is altered in a way that makes the same response easier to evoke when similar situations arise in the future. This is a form of operant conditioning, in which dopa...
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In the language used to discuss the reward system, "reward" is the attractive and motivational property of a stimulus that induces appetitive behavior (also known as approach behavior) and consummatory behavior. A rewarding stimulus is one that can induce the organism to approach it and choose to consume it. Pleasure, ...
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Within the brain, dopamine functions partly as a global reward signal. An initial dopamine response to a rewarding stimulus encodes information about the salience, value, and context of a reward. In the context of reward-related learning, dopamine also functions as a "reward prediction error" signal, that is, the degre...
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Evidence from microelectrode recordings from the brains of animals shows that dopamine neurons in the ventral tegmental area (VTA) and substantia nigra are strongly activated by a wide variety of rewarding events. These reward-responsive dopamine neurons in the VTA and substantia nigra are crucial for reward-related co...
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While dopamine has a central role in causing "wanting," associated with the appetitive or approach behavioral responses to rewarding stimuli, detailed studies have shown that dopamine cannot simply be equated with hedonic "liking" or pleasure, as reflected in the consummatory behavioral response. Dopamine neurotransmis...
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A clinical study from January 2019 that assessed the effect of a dopamine precursor (levodopa), dopamine antagonist (risperidone), and a placebo on reward responses to music – including the degree of pleasure experienced during musical chills, as measured by changes in electrodermal activity as well as subjective ratin...
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Dopamine does not cross the blood–brain barrier, so its synthesis and functions in peripheral areas are to a large degree independent of its synthesis and functions in the brain. A substantial amount of dopamine circulates in the bloodstream, but its functions there are not entirely clear. Dopamine is found in blood pl...
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The relatively small quantity of unconjugated dopamine in the bloodstream may be produced by the sympathetic nervous system, the digestive system, or possibly other organs. It may act on dopamine receptors in peripheral tissues, or be metabolized, or be converted to norepinephrine by the enzyme dopamine beta hydroxylas...
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Beyond its role in modulating blood flow, there are several peripheral systems in which dopamine circulates within a limited area and performs an exocrine or paracrine function. The peripheral systems in which dopamine plays an important role include the immune system, the kidneys and the pancreas.
https://en.wikipedia.org/wiki?curid=48548
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In the immune system dopamine acts upon receptors present on immune cells, especially lymphocytes. Dopamine can also affect immune cells in the spleen, bone marrow, and circulatory system. In addition, dopamine can be synthesized and released by immune cells themselves. The main effect of dopamine on lymphocytes is to ...
https://en.wikipedia.org/wiki?curid=48548
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The renal dopaminergic system is located in the cells of the nephron in the kidney, where all subtypes of dopamine receptors are present. Dopamine is also synthesized there, by tubule cells, and discharged into the tubular fluid. Its actions include increasing the blood supply to the kidneys, increasing the glomerular ...
https://en.wikipedia.org/wiki?curid=48548
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In the pancreas the role of dopamine is somewhat complex. The pancreas consists of two parts, an exocrine and an endocrine component. The exocrine part synthesizes and secretes digestive enzymes and other substances, including dopamine, into the small intestine. The function of this secreted dopamine after it enters th...
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The pancreatic islets make up the endocrine part of the pancreas, and synthesize and secrete hormones including insulin into the bloodstream. There is evidence that the beta cells in the islets that synthesize insulin contain dopamine receptors, and that dopamine acts to reduce the amount of insulin they release. The s...
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Dopamine as a manufactured medication is sold under the trade names Intropin, Dopastat, and Revimine, among others. It is on the World Health Organization's List of Essential Medicines. It is most commonly used as a stimulant drug in the treatment of severe low blood pressure, slow heart rate, and cardiac arrest. It is...
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Its effects, depending on dosage, include an increase in sodium excretion by the kidneys, an increase in urine output, an increase in heart rate, and an increase in blood pressure. At low doses it acts through the sympathetic nervous system to increase heart muscle contraction force and heart rate, thereby increasing c...
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Side effects of dopamine include negative effects on kidney function and irregular heartbeats. The LD, or lethal dose which is expected to prove fatal in 50% of the population, has been found to be: 59 mg/kg (mouse; administered intravenously); 95 mg/kg (mouse; administered intraperitoneally); 163 mg/kg (rat; administe...
https://en.wikipedia.org/wiki?curid=48548