Questions tagged [paradox]
A paradox is a statement or proposition that seems self-contradictory or absurd but in reality expresses a possible truth.
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Why are consecutive 6s faster conditioning on no odd rolls?
Let A be the expected number of rolls of a fair d6 until you roll two consecutive sixes, conditional on every roll up to the consecutive sixes being even.
Let B be the expected number of rolls of a ...
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Can obesity paradox among subjects with cardiovascular disease be explained by incorrectly controlling for a mediator?
Given the following causal diagram:
Obesity -> Cardiovascular disease -> Mortality
which follows from obesity being a causal factor of cardiovascular disease (CVD), isn't it reasonable to ...
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Simpson's paradox in Freedman, Pisani and Purves book
In this book, there is an example of sex bias in graduate admissions.
Major
Men
Women
# applicants
% admitted
# applicants
% admitted
A
825
62
108
82
B
560
63
25
68
C
325
37
593
34
D
417
33
375
...
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Applying Bayesian probability to a generalized Monty Hall problem
I posted this question about the Monty Hall problem and Monty's knowledge of the probability distribution several months ago. I got some good answers and this one in particular helped me gain some ...
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Odds Ratios paradox? Pooled OR inconsistent with subgroup ORs
I have two groups (A and B) that each produce ORs of 1.44 and 1.50. However, if I combine the frequencies for the two groups to create a pooled dataset, I get an OR of 1.40.
I get that it's not going ...
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A seeming paradox regarding estimation of the number of buttons
There is a computer with $N$ buttons in a secret room. We do not have access to the computer and we do not know $N$. But we know that $N\leq 100$ and we have a ever so slightly larger prior for ...
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Variation on St. Petersburg Paradox, with total loss at the end
I’m not too sure how to answer these variations I thought of and was hoping someone could enlighten me.
A casino offers a game of chance for a single player in which a fair coin is tossed at each ...
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In the Monty Hall problem, does it matter that the host knows which door the car is behind? If so, why?
If I'm thinking about this correctly, regardless of how the host chooses which door to open, there's a 1/3 chance the player initially picks the door with the car behind it, in which case they shouldn'...
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Propensity score paradox and propensity score matching
Propensity score paradox and propensity score matching
I went over the papers by King and Nielsen (2017) and Ripollone et al (2018) to figure out what is propensity score paradox in propensity score ...
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How to resolve the ambiguity in the Boy or Girl paradox?
Specifically, I was reading this article, which discusses this wording of the question:
Consider a family with two children. Given that one of the children is a boy, what is the probability that both ...
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Paradoxical Questions about confidence intervals
Let there be a repeatable real world experiment with two outcomes denoted by $0,1$ for convenience (Tossing a coin for example). Let $X_i$ be the random variable that models the ith repetition of the ...
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Is the sleeping beauty paradox really one? [duplicate]
See en.wikipedia.org/wiki/Sleeping_Beauty_problem for the statement of this problem.
This seems to me to be a simple weighted average problem. Because the total number of days awake varies between 1 (...
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How do we draw parallels between the BBG drug example and the businessman (affected by the election) anecdote in "The book of why"?
In "The book of why", Dr Pearl concludes that a BBG drug cannot exist after phrasing the sure-thing principle in 'a more correct way' (pg 214). He does this by first "insisting" ...
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Causal Inference - when Conditioning on a Collider is correct
I have been reading Judea Pearl's Book of Why and in it, he tackles the famous Monty Hall problem through a causal lens. Although it may still grind away at our initial instincts, hopefully nobody ...
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Power paradox: overestimated effect size in low-powered study, but the estimator is unbiased
If we have an underpowered study but manage to reject the null hypothesis, anyway, it makes sense to wonder if we have overestimated the effect size.
However, such a concern seems unwarranted if we ...
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simulating the St. Petersburg Paradox [duplicate]
from the following simulation it seems the fee I am willing to pay should be smaller than 10, instead of $\infty$, differs from the paradox, what is happening here?
Edit:
For those that aren't ...
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Resolve Lord's Paradox with principal components analysis (orthogonal distance regression)?
I've been reading Judea Pearl's description of Lord's Paradox in his 2018 book The Book of Why: The New Science of Cause and Effect, in which he presents the following plot:
In this special case ...
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What is actually Galton's paradox?
From One Thousand Exercises in Probability I found the exercise:
Galton's paradox. You flip three fair coins. At least two are alike, and it is an even chance that the third is a head or a tail. ...
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A variation of boy-or-girl paradox
I was working on this problem but the extra information supplied seems so superfluous and I'm trying to figure out if it is meaningful, and how it is meaningful.
Given:
This video has presented two ...
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St. Peterbourg paradox - expected value of both players
Im reading a paper by Karl Menger about the St. Peterbourg paradox (see below): his point is that the paradox does not rely on a mathematical expectation of infinite value, but on the discrepancy ...
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Question regarding asymptotic assumptions and hypotesis testing paradox for large samples
Suppose we would like to verify if a r.v $X$ follows a distribution with cumulative density as $F$, if $n$ goes to $\infty$ I'm able to use komogorov test which states reject $H_0$ (stating that $X$ ...
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Conflict between significance and importance. A paradox
When examining a set of proteins of healthy controls and patients I found significant differences using the Mann-Whitney test. However, when I used Random Forest, the most 'important' proteins for ...
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Under Berkson's Fallacy, why can't popular novels be terrible because the masses fail to appreciate quality?
Please see the embolden phrase below. Why can't lack of quality enter into or determine Popularity? I'm thinking of erotic novels which are popular because they feature lots of sex, smut, and dirty ...
user157457
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Can Lord's paradox be caused by regression to the mean?
I am trying to understand Lord's paradox, where controlling for baseline status can affect inference. I tried to set up some data following the quotation in Wikipedia
“A large university is ...
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What is the Wine/Water Paradox in Bayesian statistics, and what is its resolution?
I have just heard about the Wine/Water Paradox in Bayesian statistics, but didn't understand it very well (see Mikkelson 2004 for an introduction). Can you explain in simple terms what the paradox is ...
user314217
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Sum of individual parts not adding up to the whole
I am analyzing the WoW change in conversion rate (visitors who booked / visitors).
Let's say the WoW change from week 1 to week 2 was -10% (dropped 10%). Now, I want to know what/where this drop in ...
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Generalization of a boy or girl paradox with die!
Here is the link to the boy or girl paradox Wikipedia page. The question I have is: Let's say you have two fair 6-sided dice and you roll them simultaneously. If at least one is 6, what's the ...
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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 ...
user157457
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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 ...
user157457
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Modified sleeping beauty paradox
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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Simpson Paradox Question
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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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Average value paradox - What is this called?
I have a dataset. Say $10$ observations and $3$ variables:
...
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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 ...
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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 ...