REFUTATION OF THE LOGICAL FALLACY COMMITTED BY THE SUBJECT MATTER EXPERTS ON THE MONTY-HALL PROBLEM

MHPPIPR:©: Dr(Prof)Keshava.Prasad.Halemane Page 1 of 11 Monty-Hall-Problem PIPR:©: Dr(Prof)Keshava.Prasad.Halemane Page 1 of 11 Monty-Hall-Problem EXECUTIVE SUMMARY REFUTATION OF THE LOGICAL FALLACY COMMITTED BY THE SUBJECT MATTER EXPERTS ON THE MONTY-HALL PROBLEM Let xr{1,2,3} be the door r behind which the prize x is hidden. Let yp{1,2,3} be the initial choice p of the guest y. Let z{1,2,3} be the door q opened by the host z to show a losing choice. Also, xr and yp are mutually independent; but zq is dependent on both yp and xr, that is, zq ≠ (yp, xr). The symbol ai denotes the event [E{(a=i)}] for any ‘agent’ a{x,y,z} and ‘door’ i{r,p,q}={1,2,3}. The Table lists the 12 mutually-exclusive together-exhaustive possibilities for the combined-triple-event [xr&yp&zq] : Sl.No. [xr] [yp] [xr&yp] [zq] [xr&yp&zq] P[xr] P[yp] P[zq  (xr & yp)] P[xr&yp&zq] 01 1 1 11 2 112 1/3 1/3 1/2 1/18 02 1 1 11 3 113 1/3 1/3 1/2 1/18 03 1 2 12 3 123 1/3 1/3 1 1/9 04 1 3 13 2 132 1/3 1/3 1 1/9 05 2 1 21 3 213 1/3 1/3 1 1/9 06 2 2 22 1 221 1/3 1/3 1/2 1/18 07 2 2 22 3 223 1/3 1/3 1/2 1/18 08 2 3 23 1 231 1/3 1/3 1 1/9 09 3 1 31 2 312 1/3 1/3 1 1/9 10 3 2 32 1 321 1/3 1/3 1 1/9 11 3 3 33 1 331 1/3 1/3 1/2 1/18 12 3 3 33 2 332 1/3 1/3 1/2 1/18 Table: Twelve combined-triplet-event possibilities along with its joint-probabilities. [xr]: prize x behind door r; [yp]: guest y choses door p; [zq]: host z reveals door q Mutually-Exclusive Together-Exhaustive Alternative-Possibilities COMMENT ON THE APPROACH ADOPTED BY LEADING SUBJECT-MATTER-EXPERTS One of the various approaches adopted by the Leading Subject-Matter-Experts, is to consider the two joint probabilities P[x1&y1&z3] = 1/18 and P[x2&y2&z3] = 1/18 and compare them with the two joint probabilities P[x1&y2&z3] = 1/9 and P[x2&y1&z3] = 1/9; and somehow (wrongly) derive the probability of winning with initial choice to be 1/3 and the probability of winning with switched choice to be 2/3; without even realizing that each of these four values is an a-priori probability corresponding to the four events leading to the host revealing a losing choice; and none of them correspond to any updated a-posteriori probability that incorporates the knowledge of the losing choice revealed by the host. Also, there seems to be some Logical Fallacy in the underlying reasoning used therefor. The only one a-posteriori conditionality is (zq=z3) whereas (yp=y1) can’t be pinned down as a conditionality. Logical Consistency requires that any conditionality used in the evaluation process cannot be lifted after the evaluation process while implementing the decision arrived at based on that very conditionality. Or-Else, an erroneous problem formulation & erroneous model naturally leads to erroneous results, that gets confirmed through some erroneous computer simulation studies, etc. PROPOSED APPROACH The guest has two options, to switch or not to switch. The mathematical model must capture the central crux of the decision-making process; wherein the guest goes through a two-step procedure to arrive at the decision: (Step-1) withdraw/cancel the initial choice of door-1; (Step-2) evaluate the required a-posteriori probabilities based on the knowledge gained from the host regarding a losing choice. What is required is to compare the values of the two conditional (w.r.t. z3) marginal (w.r.t. yp) probabilities, P[x1  z3] = (P[x1  y1 & z3] + P[x1  y2 & z3]) / P[z3] = (1/18 + 1/9) / (1/3) = 1/2; and P[x2  z3] = (P[x2  y1 & z3] + P[x2  y2 & z3]) / P[z3] = (1/9 + 1/18) / (1/3) = 1/2; thus, leading to the recommendation to the guest that it really doesn’t matter either way. MHPPIPR:©: Dr(Prof)Keshava.Prasad.Halemane Page 2 of 11 Monty-Hall-Problem PIPR:©: Dr(Prof)Keshava.Prasad.Halemane Page 2 of 11 Monty-Hall-Problem REFUTATION OF THE LOGICAL FALLACY COMMITTED BY THE SUBJECT MATTER EXPERTS ON THE MONTY-HALL PROBLEM PIPR:©: Dr.(Prof.) Keshava Prasad Halemane, Professor - retired from Department of Mathematical And Computational Sciences National Institute of Technology Karnataka, Surathkal Srinivasnagar, Mangaluru - 575025, India. SASHESHA, 8-129/12 Sowjanya Road, Naigara Hills, Bikarnakatte, Kulshekar Post, Mangaluru-575005. Karnataka State, India. https://www.linkedin.com/in/keshavaprasadahalemane/ https://colab.ws/researchers/R-3D34E-09884-MI42Z https://github.com/KpH8MACS4KREC2NITK https://orcid.org/0000-0003-3483-3521 https://osf.io/xftv8/ ABSTRACT This research report presents a deep re-look at the classical Monty-Hall Problem, refuting the widely accepted position held by the Leading Subject-Matter-Experts, and establishing that there is no rational basis for a switched choice in the decision to be made by the guest of the game show. Logical consistency requires that any conditionality used in the evaluation process cannot be lifted after the evaluation process, implementing the decision arrived at based on that very conditionality. Keywords: A-Priori Probability; A-Posteriori Probability; Mutually Independent Events; Mutually Exclusive Together Exhaustive Alternatives; Joint Probability; Restricted Probability; Marginal Probability; Conditional Probability; Monty-Hall Theorem. AMS MSC Mathematics Subject Classification: 60A99; 60C99; 62A99; 62C99. 1. INTRODUCTION The classical “Monty-Hall Problem”, also referred to as the “Three-Door Problem” is based on a game show “Let’s Make a Deal” wherein the host reveals a losing choice to the guest, who had earlier made an initial choice, and in turn offers the guest an enticing option to switch from the initial choice to a second available choice with an aim to enhance the chances of winning the prize. The most prevalent & widely accepted position, as reported in literature, among the leading eminent mathematicians, statisticians, logicians, Subject-Matter-Experts and rational intellectuals, is that an appropriate detailed study & analysis of the scenario using the well accepted standard approach of Probability & Statistics, would lead to a recommendation to the guest to switch to the second available choice based on the knowledge obtained from the host revealing a losing choice. It is argued that the default of sticking to the initial choice will result in a probability of success being only one-third whereas a switch to the alternative second available choice will result in a probability of success being two-third and hence a switched choice is recommended. However, it will be shown here that this approach itself cannot be justified, and therefore the resultant recommendation for switched choice is indeed baseless. MHPPIPR:©: Dr(Prof)Keshava.Prasad.Halemane Page 3 of 11 Monty-Hall-Problem PIPR:©: Dr(Prof)Keshava.Prasad.Halemane Page 3 of 11 Monty-Hall-Problem 2. DESCRIPTION OF THE PROBLEM - BACKGROUND SCENARIO We shall focus only on the so-called classical Monty-Hall Problem, for the purpose of this report. For the sake of clarity, let us consider the standard classical Monty-Hall Problem as reported widely in the literature - with a prize hidden behind one of the three doors; a guest making a choice of the door to pick the prize; the host who knows the location of the prize as well as the choice made by the guest, now reveals a distinctly different yet a losing choice. The host also offers the guest, an option to switch from the initial choice to the now available second choice, anticipating an enhanced chance of winning the prize, based on the knowledge obtained about a losing choice. Let us represent the events/actions associated with the three doors: (1) let xr{1,2,3} be the door r behind which the prize x is hidden (2) let yp{1,2,3} be the initial choice of the door p chosen by the guest y and (3) let zq{1,2,3} be the door q opened by the host z to reveal a losing choice. The symbol ‘ai’ denotes the event [E{(a=i)}] for any ‘agent’ a{x,y,z} and ‘door’ i{r,p,q}={1,2,3}. It is essential to note here that xr and yp being mutually independent of each other as well as independent of zq, whereas zq itself is dependent on both xr and yp, as per the rules of the game. Also, note that the focus is on the decision-making process & the action to be taken by the guest. So, the problem formulation (modelling) must necessarily be from the view-point of the guest. Since the guest y has absolutely zero knowledge about xr the door r behind which the prize x is hidden, no assumptions need to be made, even about its possible probability distribution. Similarly, the initial choice yp of the door p chosen by the guest y is based on zero-knowledge without any strategy as such, and therefore at best a random (blind) guess. However, to facilitate a concrete analysis of the problem scenario and to provide a framework towards a rigorous mathematical model, it may be useful to make an assumption that these two events/actions are equally probable among the three available mutually-exclusive together-exhaustive possible alternatives, each having an a-priori probability of 1/3 thus adding up to one. Now, because the two events/actions [xr{1,2,3}] and [yp{1,2,3}] are mutually independent of each other, the joint probability of the combinations of these two events can be obtained as the product of the probabilities of the two independent component events. Therefore, the joint probability of each of the combined-duplet-events P[E{(xr{1,2,3}) & (yp{1,2,3})}] is 1/9 and the sum total of these nine joint probabilities is one. Note that the event/action of the host z opening door q, zq{1,2,3} to show a losing choice, is dependent on both yp and xr, as per the rules of the game show; that is, (zq ≠ yp) & (zq ≠ xr). Although the host has full & complete knowledge of the problem scenario, this dependency of zq on yp and xr does indeed limit his options. It turns out that when yp ≠ xr the host doesn’t have any option except to turn to the one and only one remaining door zq ≠ (yp ≠ xr); whereas when yp = xr the host has the option of choosing between the two doors, that is, zq ≠ (yp = xr). Because the host has this option, at least in a restricted sense, of choosing his door zq, it introduces an uncertainty for the guest to predict/expect/anticipate the host’s decision/action in this regard. However, as earlier, for the very same reasons as stated above, it may be useful to make an assumption that the host’s choice between the two options, whenever available, in a restricted sense, is equi-probable between the two available mutually-exclusive together-exhaustive possible alternatives, each having a restricted probability of 1/2 thus adding up to one. MHPPIPR:©: Dr(Prof)Keshava.Prasad.Halemane Page 4 of 11 Monty-Hall-Problem PIPR:©: Dr(Prof)Keshava.Prasad.Halemane Page 4 of 11 Monty-Hall-Problem 3. PROBLEM FORMULATION With the above understanding of the background scenario of the classical Monty-Hall Problem, one can derive that there are exactly 12 possibilities for the combined-triplet-event [xr&yp&zq] as represented in the Table, listing each of the 12 triplets along with the associated joint probabilities. Note that the event space is of size 12 and not 27 which would have been the case if each of the three component-events were indeed mutually independent. The first two are independent giving rise to a combined-duplet-event space E{[xr&yp]} of size nine. When it is then combined with the third component-event [._.zq], there results a splitting, in three cases. In the three cases where [xr]=[yp], that is, [x1y1.], [x2y2.], [x3y3.] the third component-event [._.zq] gets two alternative possibilities; [zq]{[z2]˅[z3]} and [zq]{[z1]˅[z3]} and [zq]{[z1]˅[z2]} respectively. Whereas in the other six cases [x1y2.], [x1y3.], [x2y1.], [x2y3.], [x3y1.], [x3y2.] where [xr]≠[yp], the third component-event [._.zq] has a fixed choice since there is one and only one single possibility satisfying the game requirement {[zq]≠([yp]≠[xr])}; no splitting into multiple alternative options. Sl.No. [xr] [yp] [xr&yp] [zq] [xr&yp&zq] P[xr] P[yp] P[zq  (xr & yp)] P[xr&yp&zq] 01 1 1 11 2 112 1/3 1/3 1/2 1/18 02 1 1 11 3 113 1/3 1/3 1/2 1/18 03 1 2 12 3 123 1/3 1/3 1 1/9 04 1 3 13 2 132 1/3 1/3 1 1/9 05 2 1 21 3 213 1/3 1/3 1 1/9 06 2 2 22 1 221 1/3 1/3 1/2 1/18 07 2 2 22 3 223 1/3 1/3 1/2 1/18 08 2 3 23 1 231 1/3 1/3 1 1/9 09 3 1 31 2 312 1/3 1/3 1 1/9 10 3 2 32 1 321 1/3 1/3 1 1/9 11 3 3 33 1 331 1/3 1/3 1/2 1/18 12 3 3 33 2 332 1/3 1/3 1/2 1/18 Table: Twelve combined-triplet-event possibilities along with its joint-probabilities. [xr]: prize x behind door r; [yp]: guest y choses door p; [zq]: host z reveals door q Mutually-Exclusive Together-Exhaustive Alternative-Possibilities 4. GENERAL ANALYSIS OF THE DECISION-MAKING SCENARIO In this section, a general analysis of the decision-making scenario (modelling, from the guest’s viewpoint) is presented first, without being constrained by the three assumptions mentioned earlier. The combined-triple-event space represented by E{[xr & yp & zq]} is partitioned into 12 mutually exclusive together exhaustive possible available alternatives – although with no assumptions about the probability distributions, just to accommodate for possible specific scenarios, especially if & when someone wishes to try out computer simulation studies, etc. Specific results pertaining to the data entries given in the Table, may always be easily worked out by plugging the corresponding data values to each of the concerned parameters as needed. The decision/choice/action of the host, represented by zq{1,2,3} being dependent on xr{1,2,3} and yp{1,2,3}; implying, that the joint probability of the combined-triplet-event referred therein be determined by the corresponding conditional probability: MHPPIPR:©: Dr(Prof)Keshava.Prasad.Halemane Page 5 of 11 Monty-Hall-Problem PIPR:©: Dr(Prof)Keshava.Prasad.Halemane Page 5 of 11 Monty-Hall-Problem That is, in general, for any [zq] and [xr] and [yp] we have, P[zq & xr & yp] = P[zq  xr & yp] * P[xr & yp]; (Eqn.1) From the rules of the game, when [zq] ≠ {[yp] ≠ [xr]} we have the conditional probability, P[zq  xr & yp] = 1; (Eqn.2) and therefore, we get the joint probability, P[zq & xr & yp] = P[xr & yp]; (Eqn.3) whereas, when [zq] ≠ {[yp] = [xr]} we cannot make any stronger statement, except the general condition for the restricted (conditional) probability: 0 ≤ P[zq  xr & yp] ≤ 1; (Eqn.4) and therefore, we get the joint probability, as per Eqn.1 above, P[zq & xr & yp] = P[zq  xr & yp] * P[xr & yp]; (Eqn.5) The entries in the Table have been filled based on the computations as in Eqn.1 to 5 above. With this information, we determine the a-posteriori (conditional on zq) marginal-probability of the prize being hidden behind door (x=r), as follows; P[xr  zq] * P[zq] = P[zq  xr & yp] * P[xr & yp] + P[zq  xr & yr] * P[xr & yr] (Eqn.6) That is, P[xr  zq] * P[zq] = P[xr & yp & zq] + P[xr & yr & zq] (Eqn.7) Note that Eqn.6 is the correct application of the Bayes-Price Rule or equivalently Eqn.7 the correct method of combining the joint probabilities to determine the marginal probability. Using Eqn.7 in specific instances, for example, with (zq=z3) as: P[x1  z3]*P[z3] = P[z3  x1 & y2] + P[z3  x1 & y1]; (Eqn.8) P[x2  z3]*P[z3] = P[z3  x2 & y1] + P[z3  x2 & y2]; (Eqn.9) Notice in Eqn.8 & Eqn.9 above, that the six terms P[x1y2z3], P[x2y1z3], P[x1y3z2], P[x3y1z2], P[x2y3z1], P[x3y2z1] are not under the control of the host, as can be confirmed from Eqn.2 & Eqn.3 above; whereas, the other six terms P[x1y1z3], P[x2y2z3], P[x1y1z2], P[x3y3z2], P[x2y2z1], P[x3y3z1] are indeed under the direct control of the host (of course, within certain limits as per Eqn.4 for the restricted probability) based on whatever strategy that the host decides and acts accordingly, while following the rules of the game. The decision of the guest as to whether to avail the offer of the host to opt for a switched choice, say, from door-1 to door-2 after knowing the losing choice behind door-3 as revealed by the host, must be based on a comparison between the two a-posteriori(conditional) marginal probabilities P[x1  z3] given by Eqn.8 and P[x2  z3] given by Eqn.9 as shown above. However, any specific answer needs to be derived based on the relative magnitudes of the four joint probabilities involved therein, namely, P[x1y1z3], P[x1y2z3], P[x2y1z3] and P[x2y2z3]. That is where the need arises to pin down certain uncertainties (at least the ones that are not under the control of the host) by assuming certain probability distribution, as for example, P[xr] and also P[yp] to be uniformly distributed among the available (in this case, three) alternatives. If the decision & action of the host can also be assumed to adhere to certain probability distribution or certain strategy, it can be used to make specific comparisons that will lead to a firm recommendation to the guest as to whether it is worth at all to consider a switched choice. The Table entries assume that the host adheres and follows a uniform distribution, in the sense that whenever faced with two/multiple alternatives, the specific choice of any one of them is equally-probable and together-exhaustive. MHPPIPR:©: Dr(Prof)Keshava.Prasad.Halemane Page 6 of 11 Monty-Hall-Problem PIPR:©: Dr(Prof)Keshava.Prasad.Halemane Page 6 of 11 Monty-Hall-Problem 5. WHAT IS WRONG WITH THE EXISTING APPROACH What is required is a comparison between the two values of the conditional (w.r.t. zq) marginal (w.r.t. yp) probabilities as can be derived using Eqn.6 & Eqn.7, or Eqn.8 & Eqn.9 above, that is, P[x1  z3] =(1/18 + 1/9)/(1/3)=1/2; and P[x2  z3] =(1/9 + 1/18)/(1/3)=1/2; (Eqn.10) thus, leading to the recommendation that it really doesn’t matter either way. One of the various approaches adopted by the Leading Subject-Matter-Experts, is to consider the two joint probabilities P113 = P[x1y1z3] = 1/18 and P223 = P[x2y2z3] = 1/18 and compare them with the two joint probabilities P123 = P[x1y2z3] = 1/9 and P213 = P[x2y1z3] = 1/9; and somehow (error of commission in the wrong application of the Bayes-Price Rule) derive the probability of winning with initial choice to be 1/3 and the probability of winning with switched choice to be 2/3; without even realizing that each of these four values is an a-priori probability corresponding to the four events leading to the host revealing a losing choice; none of them correspond to any updated a-posteriori probability that incorporates the knowledge of the losing choice revealed by the host. Also, there seems to be some Logical Fallacy in the underlying reasoning used therefor. Note that the only one a-posteriori conditionality is (zq=z3) whereas (yp=y1) can’t be pinned down as a conditionality. Logical Consistency requires that any conditionality used in the evaluation process cannot be lifted after the evaluation process while implementing the decision arrived at based on that very conditionality. Or-Else, an erroneous problem formulation & erroneous model naturally leads to erroneous results, that gets confirmed through some erroneous computer simulation studies, etc. Another approach taken by the Leading Subject-Matter-Experts seems to be based on an erroneous (error of omission) comparison between the values of the two joint probabilities – P[x1 & y1 & z3] = 1/18 and P[x2 & y1 & z3] = 1/9; (Eqn.11) which they seem to somehow wrongly combine together to get the two conditional probabilities P[x1  z3] = 1/3 and P[x2  z3] = 2/3; (Eqn.12) and then lift the conditionality (yp=y1); thus, leading to the recommendation to the guest for switching over to door-2 (that is, yp=y2). Again, note that the only one a-posteriori conditionality is (zq=z3) whereas (yp=y1) cannot be pinned down as an a-posteriori conditionality, since this very initial choice of the guest is indeed under re-evaluation and hence subject to change. It is indeed very intriguing to note that the peculiar anti-symmetry in the data as listed in the Table, P[x2y2z3] = P[x1y1z3] ≤ P[x1y2z3] = P[x2y1z3] (Eqn.13) causes the very same counter-intuitively paradoxical enticement exists for a switched choice, whatever might have been the initial choice – justifying that MHP is indeed a paradox – exactly similar to the physical analogy of a mirage, wherein the perception vanishes, since the very conditions that caused such a perception are violated by the very action of moving towards it. The mathematical model must capture the central crux of the decision-making process; wherein the guest goes through an effectively two-step procedure to arrive at the decision; (Step-1) withdraw/cancel the initial choice of door-1; followed by (Step-2) re-evaluate the required a-posteriori probabilities based on the knowledge gained from the host regarding a losing choice; and implement the decision by taking action accordingly. The clearly partitioned triple-event space, with 12 mutually-exclusive and together-exhaustive possible alternatives, represented in the Table, is a fail-safe framework to study, analyze & solve the problem – no possibility of missing any relevant (and/or including any irrelevant) component terms while going through the required calculations in order to derive whatever desired results. MHPPIPR:©: Dr(Prof)Keshava.Prasad.Halemane Page 7 of 11 Monty-Hall-Problem PIPR:©: Dr(Prof)Keshava.Prasad.Halemane Page 7 of 11 Monty-Hall-Problem 6. A CHALLENGE TO THE LEADING SUBJECT MATTER EXPERTS Let us rephrase the Monty-Hall Problem, now adorned with a jewel-on-the-crown as below: MONTY-HALL-PROBLEM (MHP) TO-SWITCH-OR-NOT-TO-SWITCH : THAT IS THE QUESTION (1.1) The prize is hidden behind one of the three doors. (1.2) I the guest make an initial choice of which door it could be, say door-1, to claim my prize. (1.3) Then Monty the host opens a different door, say door-3, revealing a losing choice. (2.1) I am given an option to withdraw/cancel the earlier choice of door-1 and switch to door-2. (2.2) I appreciate the knowledge of a losing choice and also Monty’s offer of the option to switch. (3.1) I grab Monty’s offer, withdraw/cancel my earlier choice of door-1. (3.2) Then I re-evaluate the two choices available for me now, namely door-1 or door-2. (3.3) I find that the chances of winning are exactly the same between the two available choices; (4.1) Now that YOU enter the Hall, I seek YOUR recommendation. What is YOUr recommendation? (4.2) TO SWITCH OR NOT TO SWITCH : THAT IS THE QUESTION! Note that your answer must necessarily be independent of my initial-choice (door-1); although Monty’s choice of door-3 revealing a losing choice was dependent on my initial choice (door-1) which he had to avoid as per the rules of the game. Hope your expert advice is not an exemplification of the proverb “the grass is always greener on the other side”! 7. COOL-HEADED BRAVE-HEARTS PLAY WITH STRATEGIST HOST This is somewhat far from the so-called classical version of the Monty-Hall Problem, wherein we allow the host to exercise whatever ‘strategic game-playing’ that one wishes to play with the guest. The situation can be captured by Eqn.8 & Eqn.9 above; wherein the terms P[z3  x1 & y1] in Eqn.8 and P[z3  x2 & y2] in Eqn.9 are fully under the control of the host. An extreme situation is when the host adopts whatever strategy that pulls down one of them to zero and pushes the other one to its maximum value of the restricted probability, namely 1/9. Then it turns out that the values of the two a-posteriori(conditional) marginal probabilities P[x1  z3] obtained from Eqn.8 and P[x2  z3] obtained from Eqn.9 can’t be the same anymore; in the extreme case, one will be 1/3 and the other will be 2/3; which then may lead to the two possibilities: A first specific strategy a switched choice has an clear disadvantage and a second specific strategy wherein a switched choice has an clear advantage. This topic is beyond the scope of the present paper. It is an exercise to the cool-headed brave-hearts to figure out the two specific strategies that would lead to such extreme situations. 8. CONCLUSION This research report presents a novel intriguing analysis of the Monty-Hall Problem, refuting the most widely accepted position held by the Leading-Subject-Matter-Experts - and advocating against acting on any enticing offers made by the host to the guest for an optional switch from the already selected initial choice to a distinct alternative available second choice - why, because there is no advantage gained by opting for such switched choice, in terms of any enhanced chances to win the prize, unlike whatever has been widely accepted till today. Logical consistency requires that any conditionality used in the evaluation process cannot be lifted after the evaluation process, implementing the decision arrived at based on that very conditionality, which seems to has been violated by the Leading Subject Matter Experts. MHPPIPR:©: Dr(Prof)Keshava.Prasad.Halemane Page 8 of 11 Monty-Hall-Problem PIPR:©: Dr(Prof)Keshava.Prasad.Halemane Page 8 of 11 Monty-Hall-Problem The approach taken by the Leading Subject-Matter-Experts seems to be based on an erroneous mathematical formulation of the problem, leading to an erroneous model which therefore yields erroneous results, possibly further confirmed (!?!) by erroneous computer simulation studies etc. The error can also be considered as either an error of commission, that is, a wrong application of the Bayes-Price Rule, or an error of omission, that is, considering only the aposteriori(conditionalz3)joint(x1y1z3 as against x1y2z3)probabilities rather than the required aposteriori(conditionalz3)marginal(x1y1 xor x1y2 as against x2y1 xor x2y2)probabilities. Note that any additional knowledge gained, revealing a losing (undesirable) possibility, although may lead to an updated/smaller sample-space, may not and/or need not necessarily be specific enough for a refinement/update in the relative distinction between/among the a-posteriori probabilities of the updated/now-available alternatives in the resultant updated sample-space. 9. RECOMMENDED READING [1]. Wikipedia Page - https://en.wikipedia.org/wiki/Monty_Hall_problem [2]. Jason Rosenhouse; “The Monty Hall Problem: The Remarkable Story of Math’s Most Contentious Brain Teaser”; Oxford University Press, ISBN 978-0-19-536789-8, 2009. [3]. Jason Rosenhouse; “Games-for-Your-Mind_History-&-Future-of-Logic-Puzzles”; Princeton University Press, 2020. [4]. Anthony B. Morton; “Prize insights in probability, and one goat of a recycled error”; Arxiv:1011.3400v2 2010. [5]. Matthew A. Carlton; “Pedigrees, Prizes, and Prisoners: The Misuse of Conditional Probability”; Journal of Statistics Education Volume 13, Number 2 (2005); ww2.amstat.org/publications/jse/v13n2/carlton.html [6]. Richard D. Gill; “The Monty Hall Problem is not a Probability Puzzle : It's a challenge in mathematical modelling”; arXiv:1002.0651v4 2023. [7]. Torsten Enßlin and Margret Westerkamp; “The rationality of irrationality in the Monty Hall problem”; arXiv:1804.04948v4 2018. [8]. A.P. Flitney_, D. Abbott; “Quantum version of the Monty Hall problem”; arXiv:quant-ph/0109035v3 2024. [9]. Jeffrey S. Rosenthal; “Monty Hall, Monty Fall, Monty Crawl”; probability.ca/jeff/writing_montyfall [10]. Christopher A. Pynes; “IF MONTY HALL FALLS OR CRAWLS”; EuJAP Vol.9, No.2, pp 33-47; 2013. MHPPIPR:©: Dr(Prof)Keshava.Prasad.Halemane Page 9 of 11 Monty-Hall-Problem PIPR:©: Dr(Prof)Keshava.Prasad.Halemane Page 9 of 11 Monty-Hall-Problem [11]. Andrew Vazsonyi, Feature Editor; “Which Door Has the Cadillac?”; The Real-Life Adventures of a Decision Scientist – featured column www.decisionsciences.org/DecisionLine/Vol30/30_1/vazs30_1.pdf [12]. Halemane, K.P. (2014); “Unbelievable O(L1.5) worst case computational complexity achieved by spdspds algorithm for linear programming problem”; arxiv:1405.6902 2025. 10. ACKNOWLEDGEMENT Let us acknowledge that, looking back, it seems as if Savant cast an enchantingly deep spell over Frequentists who in turn pushed Probabilists to mistake Bayes-Price, paying the price through errors of commission and/or errors of omission, riddled with Logical Fallacy. We need to come out of that long drawn intellectual hibernation of over six decades, and wake up to reality. I must necessarily confess here that the core idea behind this analysis is so stunningly & elusively simple, that one may simply be taken aback in a profound wonder-struck jaw-drop-silence, maybe with an after-thought: "oh my goodness, how could it be that it never flashed on me any time earlier"! as was also the case in an earlier research work reported in [12] by this author. On this most auspicious vidyaa(vijaya)daSami day [2025OCT02] I was blessed with the vision to formulate the Monty-Hall Theorem and its proof - as a concise & precise approach - the preferred style of presentation especially for the target audience consisting of Mathematicians, Statisticians, Logicians, and other eminent scientists etc. 11. DEDICATION To my ಅಜ್ಜ(ajja) Karinja Halemane Keshava Bhat & ಅಜ್ಜಜ(ajji) Thirumaleshwari, ಅಪ್ಪ(appa) Shama Bhat & ಅಮ್ಮ(amma) Thirumaleshwari, for their teachings through love, that quality matters more than quantity; to my wife Vijayalakshmi for her ever consistent love & support; to my daughter Sriwidya.Bharati and my twin sons Sriwidya.Ramana & Sriwidya.Prawina for their love & affection. Whereas this Original Author-Creator holds the (PIPR:©:) Perpetual Intellectual Property Rights, it is but natural that his legal heirs (three children mentioned above) may avail the same for perpetuity. To all the cool-headed brave-hearts, eagerly awaited but probably yet to be visible among the world professionals, especially the Subject-Matter-Experts, who would be attracted to and certainly capable of effectively understanding without any prejudice and appreciating the deeper insights enshrined in this short research report, who may opt for innate rational-&-intellectual common-sense and simple creativity over any sophisticated and/or complex theory in problem-solving to resolve any seemingly paradoxical scenario. 