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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 ...
Casebash's user avatar
  • 9,317
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 ...
user167920's user avatar
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 ...
Calvin Khor's user avatar
  • 35.1k
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 $b$. ...
abc's user avatar
  • 63
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 ...
aerdna91's user avatar
  • 1,142
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^{...
Christopher Wu's user avatar
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 ...
AplanisTophet's user avatar
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 ...
Idonknow's user avatar
  • 15.9k
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 ...
Mani's user avatar
  • 23
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 ...
Suraj's user avatar
  • 133
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 ...
Max's user avatar
  • 402
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 ...
Andromeda's user avatar
  • 840
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 + ...
user9576's user avatar
  • 355
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,...
axolotl's user avatar
  • 532
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^...
shark's user avatar
  • 1,011

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