All Questions
Tagged with paradoxes probability-theory
23
questions
30
votes
7
answers
5k
views
Card doubling paradox
Suppose there are two face down cards each with a positive real number and with one twice the other. Each card has value equal to its number. You are given one of the cards (with value $x$) and after ...
11
votes
5
answers
3k
views
Crisis in my understanding of probability [duplicate]
If I were to roll a die, what would be the probability of getting $2$? Certainly it would be $\dfrac 16$ (because there are $6$ numbers and sample space contains 6 numbers)
But I think we can look ...
10
votes
1
answer
975
views
How to show that the event that a prisoner does not go free is not measurable
I was reading this webpage a few months ago about the following problem-
A countable infinite number of prisoners are placed on the natural numbers, facing in the positive direction (ie, everyone can ...
6
votes
1
answer
4k
views
Expected value of the distance between 2 uniformly distributed points on circle
I have the following problem (related to Bertrand):
Given a circle of radius $a=1$. Choose 2 points randomly on the circle circumference.
Then connect these points using a line with length $b$. ...
6
votes
2
answers
514
views
Formal approach to (countable) prisoners and hats problem.
I've found this nice puzzle about AC (I'm referring to the countable infinite case, with two colors). The puzzle has been discussed before on math.SE, but I can't find any description of what is ...
4
votes
1
answer
279
views
Is there a name for this probabilistic paradox?
Let $X\sim Exp(1)$ and $Y\sim Exp(\lambda)$, independent. Then,
\begin{align}
f_{X|Y=mX}(x) = \frac{f_{X,Y}(x,mx) }{\int f_{X,Y}(x,mx) \:dx }=\frac{f_X(x)f_Y(mx) }{\int f_X(x)f_Y(mx) \:dx } = \frac{e^{...
4
votes
2
answers
90
views
Question Regarding Proposed Solution to the (Closed Envelope Version of) Two Envelope Paradox
Su, Francis, et. al. have a short description of the paradox here: https://www.math.hmc.edu/funfacts/ffiles/20001.6-8.shtml
I used that link because it concisely sets forth the paradox both in the ...
3
votes
2
answers
157
views
Set of possibilities for Simpson paradox
Italy is playing the U.S.A. in a football World Cup match. A successful pass is when a player on one team kicks the ball to a player on their team and it is not intercepted by the opposition. Is it ...
2
votes
1
answer
255
views
Is this a commonly known paradox?
I would like to know if the paradox below is commonly known and has a name.
Graham Priest, in his book Logic: A Very Short Introduction, at the end of chapter 12 “Inverse Probability“, asks the ...
2
votes
2
answers
901
views
Probability of picking a real number randomly
If we randomly pick a real number from the number line, the probability of picking a number (say x) is 0. This is true for all real numbers x and it makes sense to me why this must be true. But ...
2
votes
1
answer
115
views
A Subjective Probability Paradox When Drawing Balls From an Urn
Suppose that you are randomly drawing balls from an urn without replacement. The urn contains an unknown number of white balls and exactly one black ball. Before starting to draw, your subjective ...
2
votes
1
answer
250
views
A variation on the three prisoners problem
Three prisoners hear that one of them will be executed (the exact person who will be executed is determined upfront, and cannot be changed), while the other two will be released. Prisoner A asks the ...
1
vote
2
answers
3k
views
resolving expected utility of st. petersburg paradox with logarithmic utility
St. Petersburg paradox is a game where you toss a fair coin repeatedly and if it lands heads on the $k$th trial you get $2^n$ dollars. Expected utility of game is:
$E(U) = \sum_{k=1}^{\infty}[0.5*0 + ...
1
vote
1
answer
314
views
Conditions required to yield exactly one solution in Bertrand's Paradox
Introduction: Bertrand's Paradox
Given two concentric circles ($S_1$, $S_2$) with radii $R_1=r$ and $R_2=\frac{r}2$, what is the probability, upon choosing a chord $c$ of the circle $S_1$ at random,...
1
vote
0
answers
69
views
Lack of strong law for the Saint Petersburg paradox
The Saint Petersburg paradox can be formulated as follows: Suppose we have a lottery whose payout ${X}$ takes taking values in the powers of two ${2,2^2,2^3,\dots}$ with
$\displaystyle {\bf P}( X = 2^...