Category Archives: Probability

Test Your Intuition (18): How many balls will be left when only one color remains?

(Thanks to Itai Benjamini and Ronen Eldan.) Test (quickly) your intuition:  You have a box with n red balls and n blue balls. You take out balls one by one at random until left only with balls of the same colour. How many balls will be left (as a function of n)?

1) Roughly  εn for some ε>0.

2) Roughly \sqrt n?

3) Roughly log n?

4) Roughly a constant?

What does “beyond a reasonable doubt” practically mean?

(Motivated by two questions from Gowers’s How should mathematics be taught to non mathematicians.)

Is Backgammon in P?


The Complexity of Zero-Sum Stochastic Games with Perfect Information

Is there a polynomial time algorithm for chess?  Well, if we consider the complexity of chess in terms of the board size then it is fair to think that the answer is “no”. But if we wish to consider the complexity in terms of the number of all possible positions then it is easy to go backward over all positions and determine what is the outcome of the game when we start with each given position. 

Now, what about backgammon?  Continue reading

Emmanuel Abbe: Erdal Arıkan’s Polar Codes

Click here for the most recent polymath3 research thread. A new thread is comming soon.

Emmanuel Abbe and Erdal Arıkan

This post is authored by Emmanuel Abbe

A new class of codes, called polar codes, recently made a breakthrough in coding theory.

In his seminal work of 1948, Shannon had characterized the highest rate (speed of transmission) at which one could reliably communicate over a discrete memoryless channel (a noise model); he called this limit the capacity of the channel. However, he used a probabilistic method in his proof and left open the problem of reaching this capacity with coding schemes of manageable complexities. In the 90’s, codes were found (turbo codes and LDPC rediscovered) with promising results in that direction. However, mathematical proofs could only be provided for few specific channel cases (pretty much only for the so-called binary erasure channel). In 2008,  Erdal Arıkan at Bilkent University invented polar codes, providing a new mathematical framework to solve this problem.

Besides allowing rigorous proofs for coding theorems, an important attribute of polar codes is, in my opinion, that they bring a new perspective on how to handle randomness (beyond the channel coding problem). Indeed, after a couple of years of digestion of Arıkan’s work, it appears that there is a rather general phenomenon underneath the polar coding idea. The technique consist in applying a specific linear transform, constructed from many Kronecker products of a well-chosen small matrix, to a high-dimensional random vector (some assumptions are required on the vector distribution but let’s keep a general framework for now). The polarization phenomenon, if it occurs, then says that the transformed vector can be split into two parts (two groups of components): one of maximal randomness and one of minimal one (nearly deterministic). The polarization terminology comes from this antagonism. We will see below a specific example. But a remarkable point is that the separation procedure as well as the algorithm that reconstructs the original vector from the purely random components have low complexities (nearly linear). On the other hand, it is still an open problem to characterize mathematically if a given component belongs to the random or deterministic part. But there exist tractable algorithms to figure this out accurately.

Let us consider the simplest setting. Let X_1,...,X_n be i.i.d. Bernoulli(p) and assume that $n$ is a power of 2. Define G_n to be the matrix obtained by taking \log_2(n) Kronecker products of G_2=\begin{pmatrix}1&0\\1&1\\\end{pmatrix} , and multiply X=(X_1,...,X_n) with G_n over GF(2) to get U=(U_1,...,U_n). Note that U has same entropy as X, since G_n is invertible (its inverse is itself). However, if the entropy of X is uniformly spread out over its components, i.e., H(X)=nH(X_1), the entropy of U may not be, since its components are correlated. In any case, we can write H(U)= \sum_i H(U_i|U^{i-1}) where U^{i-1}=(U_1,...,U_{i-1}) are the `past components’. The polarization phenomenon then says that, except for a vanishing fraction of indices i (w.r. to n), each term H(U_i|U^{i-1}) tends to either 0 or 1.

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Benoît’s Fractals

Mandelbrot set

Benoît Mandelbrot passed away a few dayes ago on October 14, 2010. Since 1987, Mandelbrot was a member of the Yale’s mathematics department. This chapterette from my book “Gina says: Adventures in the Blogosphere String War”   about fractals is brought here on this sad occasion. 

A little demonstration of Mandelbrot’s impact: when you search in Google for an image for “Mandelbrot” do not get pictures of Mandelbrot himself but rather pictures of Mandelbrot’s creation. You get full pages of beautiful pictures of Mandelbrot sets


Benoit Mandelbrot (1924-2010)

Modeling physics by continuous smooth mathematical objects have led to the most remarkable achievements of science in the last centuries. The interplay between smooth geometry and stochastic processes is also a very powerfull and fruitful idea. Mandelbrot’s realization of the prominence of fractals and his works on their study can be added to this short list of major paradigms in mathematical modeling of real world phenomena.


Fractals are beautiful mathematical objects whose study goes back to the late 19th century. The Sierpiński triangle and the Koch snowflake are early examples of fractals which are constructed by simple recursive rules. Continue reading

Midrasha Talks are Now Online

Itai Benjamini listening to Gadi Kozma

There are 41 lectures from the Midrasha on Probability and Geometry: The Mathematics of Oded Schramm which are now online.

Joram Lindenstrauss’s concluding lecture (click on the picture to see)

Laci Lovasz

More pictures and links to some lectures below the line (I will slowly update them).

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