Is BlackJack Dealer Beatable?

I went to watch the recent blockbuster movie “21“, a movie based on the book “Beat the Dealer” written by Professor Edward O. Thorp. There are some apparently factual inaccuracies, such as main characters are supposed to be Asian-Americans in the MIT blackjack team, and professor Thorp wasn’t corrupt as portrayed in the movie. Plus, film directing is little disappointing. Nevertheless, there are several interesting questions mentioned in the movie.

1). Is blackjack really beatable? The answer is YES. I am not an expert on this gambling game, but I believe that it doesn’t really need some disguised signals to cheat like what Ben’s team does. Perhaps the reason is that adding little drama is just part of marketing strategy. The card game strategy like card counting is a prerequisite. There are several websites and books online dedicated to beating blackjack. Anyone who is interested in this problem may visit the website Beat the BlackJack or download the book How to Beat the Dealer in Any Casino to get acquainted with the solutions. PS: Professor Thorp also wrote an article about beating the market (allegedly) in 2003, which might interest people working on quantitative finance.

2). At the beginning of this movie, there is one scene in which Ben gives a brilliant answer to a probability problem and attracts professor Rosa’s attention. This probability problem mentioned by professor Rosa in his non-linear equation class is:

There are two goats and one car behind each of three doors. The host of the show (or game) knows exactly which door the car is behind. You are asked to pick a door first, but the host doesn’t open the door you just chose. He/she opens a door with a goat behind it. Now, you are again asked to choose one of the two remaining doors. If the car is behind the door you choose this time, then you win this car, otherwise, you will get nothing. Now the question is: What should you do? Don’t change the first choice you made or switch to the other unopened door?

Usually, the first thought is that, well, of course, you have two doors to choose, so it doesn’t matter which one you pick, the chance of winning is always 1/2. Actually, this is a simple problem that is not too simple when one is under time pressure. Ben gives a beautiful solution that looks odd at first. To see the answer in the blank space below, you need to left-click any blank space on this webpage, and then press Ctrl+A (the so-called select all command).

The answer is that changing your first choice will give your 2/3 chance of winning. Here is the interpretation. By this tactic (i.e. change the door), if the first door you chose has a goat behind it, which has 2/3 probability, then you will definitely win the car after you make the second choice. But if the thing behind the first door you chose is the car, which has 1/3 probability, then eventually you will lose after you make the second-round choice. Anyway, 2/3 chance of winning is good enough.

The 21 is an overall good entertaining movie. At least, I didn’t sleep away half of time like I did while watching Forbidden Kingdom in theater ^_^.

Filed under: Miscellaneous, The Sciences (non-technical) , 21, blackjack, card counting, Forbidden Kingdom, Life, movie, movie 21

Online Calculators

1. Online Matrix Calculator

2. Wolfram Integrator

3. Wolfram WebMathematica

4. Function Graphing Online#1 , Function Graphing Online#2

Filed under: The Sciences (non-technical)

A Tip for Showing “Page 3 of 20″ at Bottom of A Page Using LaTex

The following is a sample of codes for showing “Page 3 of 20″ (for example) at the bottom of the page. It automatically counts the number of pages and shows the current page. The package for showing the last page number is the “lastpage”.

\usepackage{fancyhdr,lastpage}

\pagestyle{fancy}
\fancyhead{}
\fancyfoot{}
\fancyhead[CE]{} %
\fancyfoot[CE,CO]{Page \thepage \ of \pageref{LastPage}}
\renewcommand{\headrulewidth}{0pt} %
\renewcommand{\footrulewidth}{0pt}

Filed under: The Sciences (non-technical)

Wolfram Demonstrations Project: Using Mathematica to Illuminate Ideas

Wolfram launches a demonstration project that uses animations to illuminate some mathematical ideas. For example, this Random Branching Process animation demonstrates a Markov process that models a population in which each branch (or individual) splits into two with a certain probability.

Filed under: The Sciences (non-technical)

Interesting Videos about Terence Tao and Grigori Perelman

YouTube has few interesting videos about these two math geniuses.

Terence Tao’s Video#1 , Terence Tao’s Video#2.

Grigori Perelman’s Video

Filed under: The Sciences (non-technical)

How to Make a Slide Using LaTex

I posted two examples of slides using beamer package of LaTex. One is the slide (PDF file) my wife made for a presentation, and here is the LaTex file for this slide. You may also be able to download another template LaTex file for the other slide (PDF file) here.

To make a PDF version of slide, one needs to first run LaTex codes using WinEdt, and convert the DVI file to PS file. Then, one may use Adobe Acrobat Professional or the website PS2PDF to convert PS file to the slide in PDF file.

Filed under: The Sciences (non-technical)

Solutions to some Problems of Preliminary Exams (2005 & 2006)

This is the response to some questions asked by some people who are working on problems of previous Statistics prelims. I am not working on statistics, and unfamiliar with statistics, either. These solutions are only based on basic understanding of some elementary statistical concepts.

I posted a solution to problem#5 of Statistics Prelim 2006 online purely using elementary calculus.

As for problem#8 of Statistics Prelim 2005, one may use Taylor expansion to directly calculate the moment generating function of variable , and it turns out to be a standard normal random variable. The calculation is straightforward combining the two conditions mentioned in this problem while writing this Taylor expansion.

To solve question c) of problem#1 of Statistics Prelim 2006, one may use the moment generating function again to prove the convergence to a normal distribution. Actually, from question b), we already know that for some constant , therefore question c) is obvious by Central Limit Theorem.

I also posted the proof for showing that the convergence in probability implies the convergence in distribution.

To prove the conclusion of Problem#6 of Stat7610 Final Exam (2005), one needs to use Chebyshev’s inequality to prove that . It’s a one-line proof.

Here I also posted a solution to Problem#1 of Stat7610 Final Exam (2006), and showed the proof of the first part. As for the second proving is asymptotically efficient, I think it’s sufficient to prove is asymptotically efficient in this case, and the calculation is pretty similar to the part showing the bias of tends to .

To show that is minimum sufficient (this is obvious) but not a complete statistic for in question a) of problem#3 of Stat7610 Midterm (2006) (You may also log on Professor Peng Zeng’s webpage for Statistics Prelims), one simply needs to prove to claim that statistic isn’t complete. As for question b), note that if were 0, then held for all . Therefore, we can prove that each -value has to be 0 by letting due to the boundedness of function .

The solution to Problem6.36 in the book Statistical Inference is also available here.

Filed under: The Sciences (non-technical)

Guide for Using LaTex to Write Math Articles for Beginners

Step1: Install the complete package of MikTex program (the latest version as of now is MikTex 2.6, and it’s totally free) online following the instructions on its webpage.

Step2: Install two most popular LaTex Compilers, WinEdt and Scientific WorkPlace. The advantage of using Scientific WorkPlace is that one can type words and math formulas on the interface similar to Microsoft Office WORD, and using shortcuts may speed up typing formulas dramatically. It also automatically saves your documents as LaTex files compatible with MikTex system. WinEdt is the most powerful and flexible (may not be the most convenient though) software for editing LaTex codes, converting LaTex files to DVI, PS or PDF files. The disadvantage of using Scientific Workplace is that it is not always compatible with MikTex system because it uses TrueTex package to edit codes, and it’s not easy to edit the header of a Latex file, either.

Step3: On my webpage for LaTex, there are several sources for LaTex study. My recommended reference book for beginners to study compiling LaTex files is The Not So Short Introduction to LATEX. You may download some sample LaTex files of different styles I wrote on my webpage from here to begin your experience on LaTex compilation.

Remark: If you are starting to write PhD dissertation, you may need to follow your school’s instruction. The instruction for writing a PhD dissertation using LaTex varies from university to university. PhD students at Auburn math department may follow the guide on how to use LaTex to write a PhD dissertation posted by Professor Tam.

Tip#1: One can also convert LaTex codes into a PDF file on the website LaTex to PDF instead of using MikTex program in your own computer.

Tip#2: The EPS image file can be previewed on the webpage Online EPS Image Viewer.

Tip#3: Ink Scape is a free software for editing EPS graphic files.

Tip#4: You may also edit your LaTex file on website ScienceSoft, and here is the list of LaTex symbols.

Filed under: The Sciences (non-technical)

A Simple Example on How to Write Math Formulas on WordPress

The codes for the formula on WordPress.com are

$ latex cov(X,Y)=\mathbb{E}\left( cov(X,Y|Z)\right) +cov(\mathbb{E}(X|Z),\mathbb{E}(Y|Z)) $

where $ latex must stick together without a space. The reason that I separated $ from the word latex is that I want to avoid that formula from being displayed again so that purely codes are shown above.

Filed under: The Sciences (non-technical)

Terence Tao’s Blog on WordPress

Here is his blog on wordpress. Sorry for that my blog’s title “coincides” with that of his.  

Filed under: The Sciences (non-technical)