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### Proving that 1- and 2-d simple symmetric random walks return to the origin with probability 1

How does one prove that a simple (steps of length $1$ in directions parallel to the axes) symmetric (each possible direction is equally likely) random walk in $1$ or $2$ dimensions returns to the ...
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### A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language

The following is a quote from Surely you're joking, Mr. Feynman . The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to ...
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### The Monty Hall problem

I was watching the movie 21 yesterday, and in the first 15 minutes or so the main character is in a classroom, being asked a "trick" question (in the sense that the teacher believes that he'll get the ...
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### Round-robin party presents (or: Graeco-Latin square with additional cycle property)

A group of $n$ people organizes recurring parties, $n-1$ events in total. At each event, each person offers a present to one other person, and each person receives a present ($n$ presents exchanged in ...
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### Which one result in mathematics has surprised you the most? [closed]

A large part of my fascination in mathematics is because of some very surprising results that I have seen there. I remember one I found very hard to swallow when I first encountered it, was what is ...
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Consider the curve $\frac{1}{x}$ where $x \geq 1$. Rotate this curve around the x-axis. One Dimension - Clearly this structure is infinitely long. Two Dimensions - Surface Area = $2\pi\int_∞^1\... 1answer 96 views ### A setting in which Rice's theorem is not true In my class we call a set of computable functions$A$recursive if its indexing set$I_A=\{e\in\mathbb N:\phi_e\in A \}$is recursive, where$\phi$is some known Gödel numbering of the computable ... 12answers 2k views ### Examples of results failing in higher dimensions A number of economists do not appreciate rigor in their usage of mathematics and I find it very discouraging. One of the examples of rigor-lacking approach are proofs done via graphs or pictures ... 1answer 191 views ### Expected number of tosses to get T,T Assume a coin has a probability p to get a head H. Suppose a coin is tossed until the partern T,T appear in the last 2 tosses. Once he got T,T then the game is finished. What is the expected number of ... 3answers 356 views ### Confusion regarding Russell's paradox Russell's paradox is about a set not in a set itself - but don't all sets are not in sets themselves?$x \in x$is not true, as {$1,2,3$}$\in${$1,2,3$} is not true.. Can anyone explain this? 1answer 245 views ### Finding an irrational not covered in standard proof that$\mu(\mathbb{Q} \cap [0,1]) = 0$[duplicate] Possible Duplicate: How would one go about proving that the rationals are not the countable intersection of open sets? Constructing a number not in$\bigcup\limits_{k=1}^{\infty} (q_k-\frac{\...
From Baire category theorem, we see that $\mathbb{Q}$ can not be a $G_{\delta}$. But consider the following construction: Let us consider $\mathbb{Q}\cap [0,1]$, putting all the elements in the set ...