Evaluating the Strengths and Limitations of Conditional Probability from the Idea of the Three-Door Problem
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Evaluating the Strengths and Limitations of Conditional Probability from the Idea of the Three-Door Problem

Yuyang Zhang 1*
1 Shenzhen College of International Education
*Corresponding author: s23207.Zhang@stu.scie.com.cn
Published on 2 October 2025
Journal Cover
TNS Vol.143
ISSN (Print): 2753-8826
ISSN (Online): 2753-8818
ISBN (Print): 978-1-80590-407-6
ISBN (Online): 978-1-80590-408-3
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Abstract

As one of the most important concepts in probability study, conditional probability plays a key role both in theoretical research and real-world applications. Derived from a famous mathematical problem—the three-door problem (also known as the Monty Hall Problem (MHP)), this paper explores the fundamental idea behind this problem with a consideration of human cognitive bias when they make a choice based on several factors. Additionally, some mathematical proofs will be included with a simulation of the MHP by using some computing skills. Apart from those theoretical concepts, this thesis also includes some evaluations of the strengths and limitations of this mathematical methods (conditionally probability) in real world situations, include medical diagnosis, risk assessment of the supply chain in a country, machine learning concept and lastly decision-making situations under some uncertainty factors which all of them have several dynamic variables that may change randomly all the time (so all the static models cannot handle any of them effectively). Lastly, some future development ideas based on current applications’ assessments are included in the conclusion part, which provide several criteria that could be further improved to address more complex scenarios and handle human special cognitive bias from their own behavior more properly in the real world in order to increase its overall accuracy.

Keywords:

Conditional probability, Monty Hall Problem, decision-making, cognitive biases.

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Zhang,Y. (2025). Evaluating the Strengths and Limitations of Conditional Probability from the Idea of the Three-Door Problem. Theoretical and Natural Science,143,24-33.

References

[1]. Oaksford, M., & Chater, N. (2003). Conditional Probability and the Cognitive Science of Conditional Reasoning. Mind & Language, 18(4), 359–379.

[2]. Borhani, F., & Green, E. J. (2018). Identifying the Occurrence or Non-Occurrence of Cognitive Bias in Situations Resembling the Monty Hall Problem. arXiv: 1802.08935.

[3]. Wilcox, J. E. (2024). Likelihood Neglect Bias and the Mental Simulations Approach: An illustration using the old and new Monty Hall problems; Judgment and Decision Making (2024), Vol. 19: e14 1–26 doi: 10.1017/jdm.2024.8

[4]. Tubau E, Aguilar-Lleyda D and Johnson ED (2015) Reasoning and choice in the Monty Hall Dilemma (MHD): implications for improving Bayesian reasoning. Front. Psychol. 6: 353. doi: 10.3389/fpsyg.2015.00353

[5]. Constantinou, A. C. (2018). Bayesian Artificial Intelligence for Decision making under Uncertainty. Engineering and Physical Sciences Research Council (EPSRC), EP/S001646/1

[6]. Tipping, M. E. (2004). Bayesian Inference: An Introduction to Principles and Practice in Machine Learning. In Advanced Lectures on Machine Learning (pp. 41–62). Springer. http: //www.miketipping.com/papers.htm

[7]. Liane Marcia Freitas Silva, Ana Camila Rodrigues de Oliveira, Maria Silene Alexandre Leite & Fernando A. S. Marins (2020): Risk assessment model using conditional probability and simulation: case study in a piped gas supply chain in Brazil, International Journal of Production Research, DOI: 10.1080/00207543.2020.1744764

[8]. Lindley, D. V. (1975). Probability and Medical Diagnosis. Journal of the Royal College of Physicians of London, 9(3), 197–204.

[9]. Juslin, P., et al. (2009). Probability Theory, Not the Very Guide of Life. Psychological Review, 116(4), 856–874. DOI: 10.1037/a0016979

[10]. Morone, A., & Adibpour, N. (2025). More Doors, More Bias? Cognitive Biases in the Extended Monty Hall Dilemma.

[11]. Gill, R. (n.d.). The Monty Hall Problem. Retrieved from http: //math.leidenuniv.nl/~gill.

[12]. Miller, J. B., & Sanjurjo, A. (2019). A Bridge from Monty Hall to the Hot Hand: The Principle of Restricted Choice. Journal of Economic Perspectives, 33(3), 144–162. doi=10.1257/jep.33.3.144

[13]. Daniele Borsetto, Mantegh Sethi, Jerry Polesel, Michele Tomasoni, Alberto Deganello, Piero Nicolai, Paolo Bossi, Cristoforo Fabbris, Gabriele Molteni, Daniele Marchioni, Margherita Tofanelli, Fiordaliso Cragnolini, Giancarlo Tirelli, Andrea Ciorba, Stefano Pelucchi, Virginia Corazzi, Pietro Canzi, Marco Benazzo, Valentina Lupato, Vittorio Giacomarra, Diego Cazzador, Luigia Bandolin, Anna Menegaldo, Giacomo Spinato, Rupert Obholzer, Jonathan Fussey & Paolo Boscolo-Rizzo (2021) The risk of recurrence in surgically treated head and neck squamous cell carcinomas: a conditional probability approach, Acta Oncologica, 60: 7, 942-947, DOI: 10.1080/0284186X.2021.1925343

[14]. Martín, J., Parra, M.I., Pizarro, M.M. and Sanjuán, E.L. (2025). A New Bayesian Method for Estimation of Value at Risk and Conditional Value at Risk. Empirical Economics, 68, 1171–1189. https: //doi.org/10.1007/s00181-024-02664-2

[15]. Saenen, L., et al. (2018). Inhibitory control in a notorious brain teaser: the Monty Hall dilemma.

[16]. Sethi, M., Borsetto, D., Cho, Y., Gair, J., Gamazo, N., Jefferies, S., Joannides, A., Mannion, R., Helmy, A., Axon, P., Donnelly, N., Tysome, J.R. and Bance, M. (2019). The Conditional Probability of Vestibular Schwannoma Growth at Different Time Points after Initial Stability on an Observational Protocol. Otology & Neurotology, 40, DOI: 10.1097/MAO.0000000000002448

[17]. Berthet V (2022) The Impact of Cognitive Biases on Professionals’Decision-Making: A Review of Four Occupational Areas. Front. Psychol. 12: 802439. doi: 10.3389/fpsyg.2021.802439

References

[1]. Oaksford, M., & Chater, N. (2003). Conditional Probability and the Cognitive Science of Conditional Reasoning. Mind & Language, 18(4), 359–379.

[2]. Borhani, F., & Green, E. J. (2018). Identifying the Occurrence or Non-Occurrence of Cognitive Bias in Situations Resembling the Monty Hall Problem. arXiv: 1802.08935.

[3]. Wilcox, J. E. (2024). Likelihood Neglect Bias and the Mental Simulations Approach: An illustration using the old and new Monty Hall problems; Judgment and Decision Making (2024), Vol. 19: e14 1–26 doi: 10.1017/jdm.2024.8

[4]. Tubau E, Aguilar-Lleyda D and Johnson ED (2015) Reasoning and choice in the Monty Hall Dilemma (MHD): implications for improving Bayesian reasoning. Front. Psychol. 6: 353. doi: 10.3389/fpsyg.2015.00353

[5]. Constantinou, A. C. (2018). Bayesian Artificial Intelligence for Decision making under Uncertainty. Engineering and Physical Sciences Research Council (EPSRC), EP/S001646/1

[6]. Tipping, M. E. (2004). Bayesian Inference: An Introduction to Principles and Practice in Machine Learning. In Advanced Lectures on Machine Learning (pp. 41–62). Springer. http: //www.miketipping.com/papers.htm

[7]. Liane Marcia Freitas Silva, Ana Camila Rodrigues de Oliveira, Maria Silene Alexandre Leite & Fernando A. S. Marins (2020): Risk assessment model using conditional probability and simulation: case study in a piped gas supply chain in Brazil, International Journal of Production Research, DOI: 10.1080/00207543.2020.1744764

[8]. Lindley, D. V. (1975). Probability and Medical Diagnosis. Journal of the Royal College of Physicians of London, 9(3), 197–204.

[9]. Juslin, P., et al. (2009). Probability Theory, Not the Very Guide of Life. Psychological Review, 116(4), 856–874. DOI: 10.1037/a0016979

[10]. Morone, A., & Adibpour, N. (2025). More Doors, More Bias? Cognitive Biases in the Extended Monty Hall Dilemma.

[11]. Gill, R. (n.d.). The Monty Hall Problem. Retrieved from http: //math.leidenuniv.nl/~gill.

[12]. Miller, J. B., & Sanjurjo, A. (2019). A Bridge from Monty Hall to the Hot Hand: The Principle of Restricted Choice. Journal of Economic Perspectives, 33(3), 144–162. doi=10.1257/jep.33.3.144

[13]. Daniele Borsetto, Mantegh Sethi, Jerry Polesel, Michele Tomasoni, Alberto Deganello, Piero Nicolai, Paolo Bossi, Cristoforo Fabbris, Gabriele Molteni, Daniele Marchioni, Margherita Tofanelli, Fiordaliso Cragnolini, Giancarlo Tirelli, Andrea Ciorba, Stefano Pelucchi, Virginia Corazzi, Pietro Canzi, Marco Benazzo, Valentina Lupato, Vittorio Giacomarra, Diego Cazzador, Luigia Bandolin, Anna Menegaldo, Giacomo Spinato, Rupert Obholzer, Jonathan Fussey & Paolo Boscolo-Rizzo (2021) The risk of recurrence in surgically treated head and neck squamous cell carcinomas: a conditional probability approach, Acta Oncologica, 60: 7, 942-947, DOI: 10.1080/0284186X.2021.1925343

[14]. Martín, J., Parra, M.I., Pizarro, M.M. and Sanjuán, E.L. (2025). A New Bayesian Method for Estimation of Value at Risk and Conditional Value at Risk. Empirical Economics, 68, 1171–1189. https: //doi.org/10.1007/s00181-024-02664-2

[15]. Saenen, L., et al. (2018). Inhibitory control in a notorious brain teaser: the Monty Hall dilemma.

[16]. Sethi, M., Borsetto, D., Cho, Y., Gair, J., Gamazo, N., Jefferies, S., Joannides, A., Mannion, R., Helmy, A., Axon, P., Donnelly, N., Tysome, J.R. and Bance, M. (2019). The Conditional Probability of Vestibular Schwannoma Growth at Different Time Points after Initial Stability on an Observational Protocol. Otology & Neurotology, 40, DOI: 10.1097/MAO.0000000000002448

[17]. Berthet V (2022) The Impact of Cognitive Biases on Professionals’Decision-Making: A Review of Four Occupational Areas. Front. Psychol. 12: 802439. doi: 10.3389/fpsyg.2021.802439

Cite this article

Zhang,Y. (2025). Evaluating the Strengths and Limitations of Conditional Probability from the Idea of the Three-Door Problem. Theoretical and Natural Science,143,24-33.

Data availability

The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.

About volume

Volume title: Proceedings of CONF-CIAP 2026 Symposium: International Conference on Atomic Magnetometer and Applications

ISBN: 978-1-80590-407-6(Print) / 978-1-80590-408-3(Online)
Editor: Marwan Omar , Jixi Lu , Mao Ye
Conference website: https://2026.confciap.org/hangzhou.html
Conference date: 30 January 2026
Series: Theoretical and Natural Science
Volume number: Vol.143
ISSN: 2753-8818(Print) / 2753-8826(Online)

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