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Imprecise probabilities alleviate the need for high-resolution and unwarranted assumptions in statistical modeling. They present an alternative strategy to reduce irreplicable findings. However, updating imprecise models requires the user to choose among alternative updating rules. Competing rules can result in incompatible inferences, and exhibit dilation, contraction and sure loss, unsettling phenomena that cannot occur with precise probabilities and the regular Bayes rule. We revisit some famous statistical paradoxes and show that the logical fallacy stems from a set of marginally plausible yet jointly incommensurable model assumptions akin to the trio of phenomena above. Discrepancies between the generalized Bayes (𝕭) rule, Dempster's (𝔇) rule and the Geometric (𝕲) rule as competing updating rules for Choquet capacities of order 2 are discussed. We note that (1) 𝕭-rule cannot contract nor induce sure loss, but is the most prone to dilation due to \"overfitting\" in a certain sense; (2) in absence of prior information, both 𝕭- and 𝕲-rules are incapable to learn from data however informative they may be; (3) 𝔇 and 𝕲-rules can mathematically contradict each other by contracting while the other dilating. These findings highlight the invaluable role of judicious judgment in handling low-resolution information, and the care that needs to be take when applying updating rules to imprecise probability models.

Forecast risk management is central to the financial management process. This study aims to apply Monte Carlo simulation to solve three classic probabilistic paradoxes and discuss their implementation in corporate financial management. The article presents Monte Carlo simulation as an advanced tool for risk management in financial management processes. This method allows for a comprehensive risk analysis of financial forecasts, making it possible to assess potential errors in cash flow forecasts and predict the value of corporate treasury growth under various future scenarios. In the investment decision-making process, Monte Carlo simulation supports the evaluation of the effectiveness of financial projects by calculating the expected net value and identifying the risks associated with investments, allowing more informed decisions to be made in project implementation. The method is used in reducing cash flow volatility, which contributes to lowering the cost of capital and increasing the value of a company. Simulation also enables more accurate liquidity planning, including forecasting cash availability and determining appropriate financial reserves based on probability distributions. Monte Carlo also supports the management of credit and interest rate risk, enabling the simulation of the impact of various economic scenarios on a company’s financial obligations. In the context of strategic planning, the method is an extension of decision tree analysis, where subsequent decisions are made based on the results of earlier ones. Creating probabilistic models based on Monte Carlo simulations makes it possible to take into account random variables and their impact on key financial management indicators, such as free cash flow (FCF). Compared to traditional methods, Monte Carlo simulation offers a more detailed and precise approach to risk analysis and decision-making, providing companies with vital information for financial management under uncertainty. This article emphasizes that the use of Monte Carlo simulation in financial management not only enhances the effectiveness of risk management, but also supports the long-term growth of corporate value. The entire process of financial management is able to move into the future based on predicting future free cash flows discounted at the cost of capital. We used both numerical and analytical methods to solve veridical paradoxes. Veridical paradoxes are a type of paradox in which the result of the analysis is counterintuitive, but turns out to be true after careful examination. This means that although the initial reasoning may lead to a wrong conclusion, a correct mathematical or logical analysis confirms the correctness of the results. An example is Monty Hall’s problem, where the intuitive answer suggests an equal probability of success, while probabilistic analysis shows that changing the decision increases the chances of winning. We used Monte Carlo simulation as the numerical method. The following analytical methods were used: conditional probability, Bayes’ rule and Bayes’ rule with multiple conditions. We solved truth-type paradoxes and discovered why the Monty Hall problem was so widely discussed in the 1990s. We differentiated Monty Hall problems using different numbers of doors and prizes.

In our world, there are many great things that defy the norm. Whether it's a marvel that takes the world by storm or a phenomenon hidden away in a humdrum form, these spectacles give us thoughts to ponder, challenge ideas that we thought were blunder, and all other things that make us Wonder.

Hintikka and Sandu's independence-friendly (IF) logic is a conservative extension of first-order logic that allows one to consider semantic games with imperfect information. In the present article, we first show how several variants of the Monty Hall problem can be modeled as semantic games for IF sentences. In the process, we extend IF logic to include semantic games with chance moves and dub this extension stochastic IF logic. Finally, we use stochastic IF logic to analyze the Sleeping Beauty problem, leading to the conclusion that the thirders are correct while identifying the main error in the halfers' argument.

We report three studies demonstrating the ‘lure of choice’ people prefer options that allow them to take further choices over those that do not, even when the extra choices cannot improve the ultimate outcome. In Studies 1 and 2, participants chose between two options: one solitary item, and a pair of items between which they would then make a further choice. Consistent with the lure of choice, a given item was more likely to be the ultimate choice when it was initially part of a choice pair than when it was offered on its own. We also demonstrate the lure of choice in a four‐door version of the Monty Hall problem, in which participants could either stick with their original choice or switch to one of two unopened doors. Participants were more likely to switch if they could first ‘choose to choose’ between the two unopened doors (without immediately specifying which) than if they had to choose one door straightaway. We conclude by suggesting that the lure of choice is due to a choice heuristic that is very reliable in the natural world, but much less so in a world created by marketers. Copyright © 2003 John Wiley & Sons, Ltd.

The application of probabilistic arguments to rational decisions in a single case is a contentious philosophical issue which arises in various contexts. Some authors (e. g. Horgan, Philos Pap 24: 209-222, 1995; Levy, Synthese 158: 139-151, 2007) affirm the normative force of probabilistic arguments in single cases while others (Baumann, Am Philos Q 42: 71-79, 2005; Synthese 162: 265-273, 2008) deny it. I demonstrate that both sides do not give convincing arguments for their case and propose a new account of the relationship between probabilistic reasoning and rational decisions. In particular, I elaborate a flaw in Baumann's reductio of rational single-case decisions in a modified Monty Hall Problem.

Probabilities or risks may change when new information is available. Common sense frequently fails in assessing this change. In such cases, Bayes' theorem may be applied. It is easy to derive and has abundant applications in biology and medicine. Some examples of the application of Bayes' theorem are presented here, such as carrier risk estimation in X-chromosomal disorders, maximal manifestation probability of a dominant trait with unknown penetrance, combination of genetic and non-genetic information, and linkage analysis. The presentation addresses the non-specialist who asks for valid and consistent explanations. The conclusion to be drawn is that Bayes' theorem is an accessible and helpful tool for probability calculations in genetics.

Probability is notorious for problems that are easy to state but whose solutions can lead to confusion, heated disputes, and hurling of insults. Most often the confusion stems from problems that are not well formulated and several different interpretations are possible. The high emotional level may have to do with the fact that probability problems often relate to everyday phenomena. The “Monty Hall problem” is probably the most famous of all probability problems. It created a drama that involved a TVshow, the world's smartest person, several rude mathematicians, one car, two goats, and countless lunch discussions all over the world. In many problems in probability, it is difficult to have an intuitive grasp of what the correct answer should be, even if the problem is simple to state and refers to something familiar. Probability problems are in this way different from other types of everyday math problems.

As a purely mental phenomenon, probability is completely free‐floating, not intrinsically tethered to the real world in any sense. It is useless if it is completely divorced from reality. Fisher's view of probability was first stated in one of his earliest papers as the probability is something that is “imagined” and refers to an “infinite hypothetical population” of objects. Fisher attempted high‐wire act of maintaining Jacob Bernoulli's balance on razor's edge. His exposition of probability as essentially metaphorical was completely out of step with his times. Fisher, like Keynes, understood that determining a reference class from which the case in hand could be regarded as random representative was fundamental. He articulated three requirements that are necessary for a valid statement of mathematical probability. In a similar way, knowing true protocol in Monty Hall Problem would allow the player to determine whether or not Monty's clue was relevant in Fisher's sense.