A paradox is a statement or proposition that seems self-contradictory or absurd but in reality expresses a possible truth.

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### Borel Cantelli conceptual problem

I recently learned about the Borel-Cantelli lemmas and although I could follow the derivation and it seems watertight, I have a problem wrapping my head around its implication. I hope you can tell me ...
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### Abraham Wald survivorship bias intuition

During World War II, the statistician Abraham Wald took survivorship bias into his calculations when considering how to minimize bomber losses to enemy fire. Wald noted that the study only considered ...
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### Extremely Long Runs in Bernoulli Process

I caught my son counting his ribs during a biology exam. As punishment for this act of cheating, I set him in the corner with a fair coin and told him he must stay in the corner, flipping the coin, ...
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### Berkson's Paradox: How does further conditioning increase the negative correlation?

I don't understand the bolded sentence beneath from Cory Simon's BSc Ohio, PhD UBC old blog. How's negative correlation in the red triangle $<$ (to wit, 'more severe') negative correlation in ...
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### How can the nicest men in the conditioned subset be as nice as the average person in the whole population?

I don't grasp the bolded sentence beneath from Jordan Ellenberg's article. His diagram no longer renders, so I use these. How can the nicest men in the green triangle be as nice as the whole ...
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Consider the following classic problem: Some researchers would like to put Sleeping Beauty to sleep on Sunday. Depending on the secret toss of a fair coin, they will briefly awaken her either once ...
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### Bayesian comparison of differing model size

This issue is related to a real problem, but I've boiled down to a minimal example. Simplest version Suppose I get to observe two variables, $X_1$ and $X_2$. We can think of these data being ...
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### A reliance on repeated sampling ideas can lead to logical paradoxes that appear in common rather than esoteric procedures?

I am currently studying the textbook In All Likelihood -- Statistical Modelling and Inference Using Likelihood by Yudi Pawitan. Section Repeated sampling principle: the frequentists of chapter 1 says ...
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### Paradox game for life and death

This is probably not new to any student of statistics - yet I can't find the name of that "paradox" to search for. Imagine the richest person on earth holds a game: A participant is invited to roll ...
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I am trying to understand if the following statement is an example for Simpson paradox: "In the US elections a certain candidate got more votes than the other, but the other one was elected". I ...
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### Famous easy to understand examples of a confounding variable invalidating a study

Are there any well-known statistical studies that were originally published and thought to be valid, but later had to be thrown out due to a confounding variable that wasn't taken into account? I'm ...
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### Probabilities in the Raven paradox

The raven paradox is roughly: "The statement All ravens are black is logically equivalent to All non-black entities are not ravens. Whenever we observe a non-black non-raven, the probability for the ...
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### Did Jaynes ever comment on Lindley’s paradox?

I wondered whether ET Jaynes ever wrote or expressed an opinion about Lindley’s famous statistical paradox? I would be curious about his take on it, and imagine he must have done since he wrote ...
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### Twist to 3 prisoners problem applying Bayes rule

T, J and B work for a company but the chairman has decided to fire one person randomly chosen through 3 cards. The chairman decides to fire with unequal probabilities -- T with probability of 15%, B ...
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I don't understand how the following [I reworded the original] instances Berkson's Paradox; and can be analogized to the University Admissions example, where $\color{limegreen}{\text{corr(SAT, GPA)}&... 3answers 2k views ### How to program a Monte Carlo simulation of Bertrand's box paradox? The following problem has been posted on Mensa International Facebook Page:$\quad\quad\quad\quad\quad\quad\quad\quad$The post itself received 1000+ comments but I won't go into details about the ... 1answer 66 views ### What is the name of the statistical paradox / law that says optimizing the metric may not lead to optimal outcome I am trying remember to the name of the statistical paradox / law name that effectively says that optimizing the metric may not lead to the optimal outcome. As far as I can recall, this paradox / law ... 21answers 25k views ### At each step of a limiting infinite process, put 10 balls in an urn and remove one at random. How many balls are left? The question (slightly modified) goes as follows and if you have never encountered it before you can check it in example 6a, chapter 2, of Sheldon Ross' A First Course in Probability: Suppose that ... 2answers 511 views ### False positive paradox, the definition of “accuracy” I have a question about the false positive paradox. It is an example that is often used to motivate Baye's formula. I first give an explanation of the paradox. Suppose there is a device which is 90 ... 0answers 422 views ### Which model is better based on test and training accuracy I have this assignment question: You are given a dataset for cancer detection having two classes (binary classification). 0 stands for “cancer not detected” and 1 for “cancer detected”. This ... 2answers 280 views ### T-test paradox: can adding a single point very far from the null value change the outcome from significant to nonsignificant? Let's suppose we have the situation presented in picture$1$. We have a set of$n \in \mathbb{N}$points that have a mean larger than null hypothesis:$\bar{x}>x_{H_0}$. Also$var(x)$is such that ... 6answers 4k views ### Average value paradox - What is this called? I have a dataset. Say$10$observations and$3$variables: ... 3answers 554 views ### Paradox of Poisson process with at least one event in the interval Let$X_T$is a number of events in Poisson process of unitary rate ($\lambda = 1$) within interval of length$T$. It is known that at least one event has been observed in the interval, I want to find ... 0answers 164 views ### Real world examples of the sleeping beauty paradox The Sleeping Beauty problem is a thought experiment concerning a participant, Sleeping Beauty, who is woken once or twice based on the flip of coin and is asked her degree of belief on the coin having ... 3answers 2k views ### What's wrong with this proposed resolution to the St Petersburg Paradox? We have a game where your payout is$2^k$where$k$is the number of times you flipped a coin to land on heads (if your first flip is a head, then$k=1$). Then the expected payout is:$$E = \frac{1}{2}... 2answers 223 views ### How can A and B have equal chances to be visited last if A has higher chances to be visited before B? The classical problem considered by Ross is a random particle visiting all chairs under a circular table. That is, there is a circular buffer of size m+1, were ... 3answers 616 views ### What is the name of this statistical paradox/phenomenon: statistical significance depends on what would have happened but did not I don't remember it exactly, and I'd like to read up on it again. It is about experiment design and frequentist hypothesis testing. A sequence of N measurements are made, and then when evaluating the ... 4answers 2k views ### Blackwell's bet I have read about Blackwell's bet paradox on Futility closet. Here is the summary: you are presented with two envelopes,$E_x$and$E_y$. The envelopes contain a random amount of money, but you don't ... 1answer 66 views ### What Are Some Historical Applied Stats Problems [closed] What are some interesting, historical applied statistics problems ? i.e. sometime in the past that statistics have been used to affect the course of events? The kind of problems I am looking for ... 3answers 4k views ### Paradox in model selection (AIC, BIC, to explain or to predict?) Having read Galit Shmueli's "To Explain or to Predict" (2010) and some literature on model selection using AIC and BIC, I am puzzled by an apparent contradiction. There are three premises, AIC- ... 0answers 51 views ### What basis should be used to devize a significance/hypothesis test? I am trying to understand the basis of hypothesis testing and I came up with a paradox. Let me explain with an example. Consider the case of trying to determine whether is coin is fair. The null ... 1answer 92 views ### Probability to be the millionth customer (What Would You Do?) I saw this episode of "What Would You Do?" a few months ago, and I keep wondering what would statistically be the best thing to do in this situation. Here is the problem formulation: You are waiting ... 1answer 386 views ### Does Stein's Paradox still hold when using the$l_1$norm instead of the$l_2\$ norm?

Stein's Paradox shows that when three or more parameters are estimated simultaneously, there exist combined estimators more accurate on average (that is, having lower expected mean squared error) than ...
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After seeing this question, I thought I would try to simulate the bus waiting time paradox to help my understanding. However, what I got was the "intuitive" result, rather than that predicted by the ...
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A few years ago I designed a radiation detector that works by measuring the interval between events rather than counting them. My assumption was, that when measuring non-contiguous samples, on ...
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### Simpson's Paradox with new sample

Suppose you constructing model whose training data is cumulative in nature; meaning each year you can add new observations with all prior observations being kept the same. (e.g. training set is non-...
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### Unkown 6-sided dice. After 600 rolls frequency for all sides exactly equal. What is the chance, that rolling “6” with this dice has frequency > 1/6?

Although it is unknown dice, the symmetry of the evidence tells us, that we can treat the dice as fair, so the chance should be exactly 50%. But if we simulate it by hand, the result is less then 50%:...
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### Two envelope problem revisited

I was thinking of this problem. http://en.wikipedia.org/wiki/Two_envelopes_problem I believe the solution and I think I understand it, but if I take the following approach I'm completely confused. ...
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### A seeming paradox with rational agents not coming to the same conclusion given the same data

So one day after a tasty dinner full of bananas, an idea comes to your mind (you are person A) - "What if eating bananas can cure cancer?". Being a scientist at heart, you conduct a double blind study ...
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### Possible Paradox: Calculating a confidence interval with within-experiment error

This is a spinoff of How to calculate the confidence interval of the mean of means? and related to When making inferences about group means, are credible Intervals sensitive to within-subject ...
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### Name of the “paradox” reported by Gelman

In Andrew Gelman's book "Red State, Blue State" he analyzes the fact that rich people within particular states tend to vote more Republican than poor people, but that wealthy states tend to vote more ...
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Model The firm and enforcement agency interact in more than one domain. This may arise because a single agency is responsible for enforcing more than one regulation or because it enforces the same ...
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### Swapping X and Y in a regression that contains a grouping predictor?

Suppose I'm doing a linear regression and I want to investigate how the association between a predictor X and a response Y changes according to levels of a 2-level factor G. The model would look like ...
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The situation Some researchers would like to put you to sleep. Depending on the secret toss of a fair coin, they will briefly awaken you either once (Heads) or twice (Tails). After each waking, they ...
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### Does the principle of indifference apply to the Borel-Kolmogorov paradox?

Consider Jaynes' solution to the Bertrand paradox using the principle of indifference. Why doesn't a similar argument apply to the Borel-Kolmogorov paradox? Is there something wrong with arguing that ...
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### What will be the correct answer, if we modify the “Best statistics question ever”?

There is a popular question, called "Best statistics question ever". If you choose an answer to this question at random, what is the chance you will be correct? A) 25% B) 50% C) 60% D) 25% This ...