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Showing posts from July, 2016

### Math Puzzle: Rich Get Richer?

Today I'm going to talk about a math problem whose answer is simultaneously simple and unintuitive.  Here is the problem statement:
You have a urn that has $1$ red ball and $1$ blue ball.  You repeatedly remove one ball from the urn randomly, then put two balls of the same color back into the urn.  How does this system behave over time?  Specifically, what is the probability that at some point the urn will contain exactly $r$ red balls and $b$ blue balls? If you want to see my solution, go ahead and expand the solution tab.  Otherwise, you can try to figure it out for yourself.  Also, if you have other interesting questions you want to ask about this system, feel free to post them in the comments below.

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Intuitively, this seems like a "rich get richer" problem", in that the long term behavior of the system depends on how "lucky" we are in the early stages (i.e., how many red and blue balls we pick).  For example, if we pick mostly red ball…

### Game Theory, Group Projects, and Google PageRank

One of the most common complaints about group projects in college is about work distribution: some people put in more work than others, and the people who put in more work do not think that is fair.  I've worked with many different groups in college, and I've been on both sides depending on the class and the project.  However, I don't actually mind putting in more work than other people when I enjoy the class and I think the project will provide a valuable learning experience.  By taking the lead, I end up understanding the project material much better than I would have otherwise.  However, when I don't care about the class as much, I tend to do less work.  So while I can't necessarily relate to other people who put in more work and complain about their teammates, I will offer some explanation as to why some people sit on the sidelines while other people take on the burden of doing the majority of the project.

The first observation to make is that not everybody ha…

### Beat the Streak: Day Six

In this blog post, I am going to show why the work I did on Day Three is so important, and how using the strategy I outlined in that post can improve your odds of beating the streak by a factor of 5-10!  On day three, I analyzed the situations under which you should select a player who you think will get a hit, as opposed to not selecting that player, and instead maintaining your current streak until the next day.  In summary, I found that your decision should be guided by your current streak, the number of games left in the season, your confidence in the player (how likely is he to get a hit?), and the distribution of likelihoods across all games in the season.  After solving for the optimal strategy, I was able to approximate the probability of winning under that strategy by making some simplifying assumptions about the probability distribution of the best player getting a hit on a given day.

I'm not going to get too deep into the math in this blog post, but I want to give you …

### Challenge Problem: Taking a Quiz

I thought of this problem about two years ago and was going to send it out as the one of the weekly challenge problems for the ACM student chapter at UD, however I was unable to solve it when I first thought of it, and so I put it on the back-burner for a while.  Recently, I revisited it and made some progress on it, but was still unable to find a strategy that can solve it at all, let alone efficiently.  I am posting the problem to this blog in the hopes that you can help me figure it out by posting your ideas and code in the comments below.  Anyway, here is the problem:
You are taking a quiz with $n$ true/false questions that you didn't study for, but you are allowed to retake the quiz as many times as you want. Every time you take the quiz, you are told how many questions you got correct, but not which questions were right and wrong.  Your final score is the number of answers you got correct on your last submission minus the number of times you retook the quiz.  Create a progra…

### New Website

Today I finished porting the content of my blog to this location.  It was a tedious process, but I think this is a more user friendly blogging platform than what I was using before, so it will pay off in the long run.  Blogger is a nice platform because they can host your blog for free, so the only thing I have to pay for to maintain this site is \$12/year for the domain.  I'm not a web development expert, and the drag-and-drop nature of Blogger will allow me to produce content a little bit faster, and possibly nicer.

Anyway, I decided to decouple my old website into two components: (1) a personal website which contains mostly static content and is more professional, and (2) this blog, which is more dynamic and informal.  However, there were some aspects of my old website that don't easily fit into either category, such as the pages about my academic interests, so I am going to slowly bring some that content into this blog and discard the rest of it.

Here's a few things tha…