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Taking free online courses is gaining momentum. There are many options to choose from: Khan Academy, Education – YouTube, and Udacity. Some come with ivy league and other top universities backing such as MIT’s and Harvard’s edX (a Joint venture builds on MITx and Harvard distance learning), and Coursera which offers host courses from Princeton University, Stanford University, University of California, Berkeley, University of Michigan-Ann Arbor, and University of Pennsylvania.

The world is watching and people have welcomed this movement with open arms! For instance, Udacity’s (founded by three roboticists) first class, “Introduction to Artificial Intelligence,” over 160,000 students in more than 190 countries enrolled! They aim to continue this trend by teaching 160,000 plus students statistics with it’s latest course: “Intro to Statistics (ST101) Making Decisions Based on Data.” This class is being taught by leading computer scientist and Udacity co-founder Sebastian Thurn a PhD in computer science and statistics.

Khan Academy is not-for-profit with the goal of changing education for the better by providing a free world-class education to anyone anywhere.

Coursera is a social entrepreneurship company that partners with the top universities in the world to offer courses online for anyone to take, for free.

Udacity is a private educational organization that believes university-level education can be both high quality and low cost (For some courses, you will have an opportunity to take a certified exam, which will have some cost.)

Although it happened about a month ago, I’ve only just heard that John Ball can be added to the list of current and former Presidents of the International Mathematical Union who have signed the Cost of Knowledge boycott. He had to resign from editorships of three Elsevier journals to do so.
While I’m on this kind of topic, there are under two days left to sign the White House open access petition. Although it’s reached 25,000 signatures and will therefore get an official response, it is still the case that the more signatures it gets the better.

Here are the two relevant links.

https://wwws.whitehouse.gov/petitions#!/petition/require-free-access-over-internet-scientific-journal-articles-arising-taxpayer-funded-research/wDX82FLQ

http://thecostofknowledge.com/

“Rounding is more difficult than first appears. It appears straight-forward. To round a number you decide how many decimal places or significant figures you need then you look one digit further to see whether the final digit stays the same or goes up. Presto – there is rounding in a nutshell. Yet my university students struggle with rounding to a surprising degree. I did a Youtube search on rounding for a video to help them, but to no avail.

I wrote a script for such a video. I’m afraid it won’t be appearing any time soon as I now have to work for my living (as opposed to being an academic ) but the exercise was interesting. What I realised is that rounding is about communication. It has nothing to do with mathematics and everything to do with expression.”

Two quick updates with regards to polymath projects.  Firstly, given the poll on starting the mini-polymath4 project, I will start the project at Thu July 12 2012 UTC 22:00.  As usual, the main research thread on this project will be held at the polymath blog, with the discussion thread hosted separately on this blog.

Second, the Polymath7 project, which seeks to establish the “hot spots conjecture” for acute-angled triangles, has made a fair amount of progress so far; for instance, the first part of the conjecture (asserting that the second Neumann eigenfunction of an acute non-equilateral triangle is simple) is now solved, and the second part (asserting that the “hot spots” (i.e. extrema) of that second eigenfunction lie on the boundary of the triangle) has been solved in a number of special cases (such as the isosceles case).  It’s been quite an active discussion…

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I recently experienced three consecutive days that are major anniversaries for me: June 12 is my baptismal anniversary, June 13 is my birthday, and June 14 is my wedding anniversary (also my daughter turned 15 months old on this day). These events put to the forefront of my mind a famous problem in probability, the birthday problem.

— The Uniform Probability Model for a Finite Sample Space —

There are many different probability models. We’ll start with the simplest of these models, the uniform probability model for a finite sample space.

Let ${S}$ be a finite sample space. The uniform probability model ${P}$ for ${S}$ is defined by

$\displaystyle P(E)=\frac{|E|}{|S|}.$

${P}$ as defined satisfies the axioms for a probability measure. By adopting this model, we are predicting that in a sequence of experiments with sample space ${S}$ the observed relative frequency of events${E}$ in the sequence of experiments will be equal to the relative frequency, ${|E|/|S|}$, of events ${E}$ in the set ${S}$. In particular, for each outcome ${s\in S}$, we have ${P(\{s\})=1/|S|}$ , or as one says, “All outcomes in ${S}$ are equally likely (i.e., equally probable)." Incidentally, for events ${E=\{s\}}$ consisting of a single outcome are sometimes called elementary events.

Problem 1 (The birthday problem) In a room of ${n}$ people, what is the probability ${p_{n}}$ that at least two have the same birthday (i.e., same month and day of the year)? Find the smallest ${n}$ such that ${p_{n}\geq\frac{1}{2}}$. Neglect February 29 and assume a year of ${365}$ days.

Solution: Let ${S}$ be the sample space that consists of ${365^{n}}$ ${n-}$tuples ${(b_{1},b_{2},\ldots,b_{n})}$, where ${b_{i}}$ is birthday of the ${i^{th}}$ person in the room. Note, that the probability is

$\displaystyle \begin{array}{rcl} p_{n} & = & 1-P(\mbox{all n individuals have different birthdays})\\ & = & 1-\frac{365{}^{\underline{n}}}{365^{n}}, \end{array}$

assuming all ${n-}$tuples ${(b_{1},b_{2},\ldots,b_{n})}$ are equally likely. Below are some values of $p_{n}$:

$\begin{array}{|c|c|} p & p_{n} \\ \hline 5 & 0.027 \\ 10 & 0.117 \\ 20 & 0.411 \\ 23 & 0.507 \\ 30 & 0.706 \\ 40 & 0.891 \\ 60 & 0.994 \\ \hline \end{array}$

Remarkably, in a room with just ${23}$ people, the probability is greater than ${1/2}$ that two or more people have the same birthday! $\Box$

— Binomial Random Variables and Poisson Approximations —

In the case of a binomial experiment there are ${n}$ independent trials, with the results of each trial labeled “success" or “failure." On each trial the probability of success is ${p}$ and (hence) the probability of failure is ${1-p}$. The sample space ${S}$ consists of the set of ${2^{n}}$ words in the alphabet ${\{s,f\}}$ and the random variable ${X}$ on ${S}$ is defined by ${X}$ (each word) ${=}$ the number of ${s}$‘s in that word. The probablity density function (pdf) ${f_{X}}$ for ${k=0,1,\ldots,n}$ is given by

$\displaystyle f_{X}(k)=P(X=k)=\binom{n}{k}p^{k}(1-p)^{n-k}.$

We say that such a random variable ${X}$ is a binomial random variable with parameters ${n}$ and ${p}$, abbreviating this with the notation ${X\sim binomial(n,p)}$. We also say that X has a binomial distribution with parameters ${n}$ and ${p}$, and that ${f_{X}}$ is a binomial density function with parameters ${n}$ and ${p}$. Note, in general, if ${X\sim binomial(n,p)}$, then ${P(X\geq1)=\sum_{k=1}^{n}P(X=k)=1-P(X=0)=(1-p)^{n}}$.

If ${X\sim binomial(n,p)}$, where ${n}$ is “large" and ${p}$ is “small," then for ${k=0,\ldots,n}$

$\displaystyle P(X=k)\approx\frac{\lambda^{k}}{k!}e^{-\lambda},$

where ${\lambda=np}$. This is called the Poisson approximation (Denis Poisson, 1781-1840) to the binomial distribution.

Remark 1 It can in fact be shown that the accuracy of this approximation depends largely on the value of ${p}$, and hardly at all on the value of ${n}$. The errors in using this approximation are of the same order of magnitude as ${p}$, roughly speaking.

Problem 2 Given ${400}$ people, estimate the probability that 3 or more will have a birthday on June 13.

Solution: Assuming a year of ${365}$ days, each equally likely to be the birthday of a randomly chosen individual, if ${X}$ denotes the number of people with a birthday on June 13 among ${400}$ randomly chosen individuals, then ${X\sim binomial(400,\frac{1}{365})}$. The exact answer to this question

$\displaystyle \begin{array}{rcl} P(X\geq3)& = & 1-P(X\leq2)\\ & = & 1-\sum_{k=0}^{2}\binom{400}{k}\left(\frac{1}{365}\right)^{k}\left(\frac{364}{365}\right){}^{400-k}\\ & = & 0.09850825486213655 \end{array}$

The Poisson approximation of this quantity is

$\displaystyle \begin{array}{rcl} 1-\sum_{k=0}^{2}\frac{(400/365)^{k}}{k!}e^{-400/365} & = & 1-e^{-1.096}(1+1.096+\frac{(1.096)^{2}}{2})\\ & \approx & 0.099. \end{array}$ $\Box$

A post from one of the blogs I follow, which happens to combine my love of mathematics and programming. Check it out! 🙂

After a year of writing this blog, what have I learned about the nature of the relationship between computer programs and mathematics? Here are a few notes that sum up my thoughts, roughly in order of how strongly I agree with them. I’d love to hear your thoughts in the comments.

1. Programming is absolutely great for exploring questions and automating tasks. Mathematics is absolutely great for distilling the soul of a problem.
2. Programming is fueled by the excitement of what can be done. Mathematics is fueled by the excitement of how things relate, and why they relate.
3. Good mathematics makes for short programs.
4. Good mathematics can be sloppy. Good programs cannot.
5. Useful algorithms can come from any branch of mathematics, so it is best to be familiar with them all (at least a little).
6. Most programs written for the real world use no mathematics beyond the level of an average twelve-year-old, but

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Chris Evans has proposed a new polymath project, namely to attack the “Hot Spots conjecture” for acute-angled triangles.   The details and motivation of this project can be found at the above link, but this blog post can serve as a place to discuss the problem (and, if the discussion takes off, to start organising a more formal polymath project around it).

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Today, June 2, 2012, the number of signatures have reached 24,000! Only 1000 signatures are needed to reach the target before June 19, 2012.

Make it happen! Sign the petition here.

I enjoyed this blog post, so I am sharing it. 🙂 Check it out.

I know a fair bit about the probability of winning the lottery.

State-run lotteries are a tax on the mathematically challenged.

Given the odds of winning, then you might wonder why I occasionally buy scratch-off lottery tickets. Lord knows, my wife often wonders aloud about it. Believe it or not, there are three reasons that I buy these tickets:

• First, I’m from a rural town in the-middle-of-nowhere Pennsylvania. The rural poor are infamous consumers of lottery tickets. Consequently, I believe that buying lottery tickets is part of my genetic code.
• Second, it’s a guilty pleasure that is easier to indulge than buying PowerBall or Daily Number tickets. When you buy one of those, there is a human interaction, and I imagine that the clerk selling me the ticket is thinking, “Loser! Don’t you know how low your odds of winning are?” For the scratch-off tickets, you insert your money in a…

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