The Birthday Problem asks the question: In a group of n people, what's the probability that at least 2 will have the same birthday?
Answering the Birthday Problem involves creating a mathematical model. The model rests on two assumptions that aren't true and should be discussed with the class:
- there are 365 days in a year (Feb 29th is ignored to simplify the model)
- birthdays are equally likely to be on any given day (Also false. This varies from country to country; in the US birthdays are more towards the middle of the year. Count back 9 months and you've got cold weather. Nothing random there.)
Under those assumptions the sample space is , since there are 365 choices for each birthday. Counting the number of ways that nobody has the same birthday is just (365)(364)....(365-n+1), giving us the probability of at least 2 people with the same birthday as the complementary probability: 1-[(365)(364)....(365-n+1)]/(365)^n$.
This simple model yields surprising answers: with a group of only 50 people the probability of at least 2 people with the same birthday is 0.97; this illustrates how counter-intuitive probability can be. So is the model any good? It turns out that although the model is built on incorrect assumptions, experimental evidence validates gives results that are close to the model. That's a basic fact about models; they're based on assumptions that in many cases aren't true. Ultimately, experimental evidence is needed to determine whether the model is accurate. Try the Birthday Problem with each of your classes and find out how well it works with 1 class, with any 2 classes, any 3 classes, and so on.
I've created code for computing the Birthday Problem given n people and plots the probabilities on a graph. You can see the output above; the code is on the Python/Sage page.
Some stories that caught my eye recently:
- Education Next has articles on "Rethinking the High School Diploma". Three articles consider the idea of having 2 diplomas: 1 for graduating and the second higher diploma for excellence/mastery. It seems symptomatic of today's culture. You can't enforce standards so everybody needs to graduate and the diploma lacks value. So create a second diploma which will show the employer the student has mastered the basics. An interesting idea.
- Businessweek has an article suggesting that the US should station soldiers in schools, similar to air marshalls on a plane. The Call of Duty author has, "...anticipated objections. “The public won’t like it, they’ll think it’s a police state,” he said. But, he went on, “All of these are solvable problems.” Anthony’s address, which was punctuated by videos depicting such future threats as a U.S. drone hacked by Iran and a hotel massacre in Las Vegas, included repeated exhortations to policymakers to learn from the examples of corporations and creative artists in selling potentially unpopular ideas. “When we have a new product that has elements that we’re not sure how people will respond to, what do we do as a corporation?” he asked. “We market it, and we market it as much as we can—so that whether people like it or not, we do all the things we can to essentially brainwash people into liking it before it actually comes out.”". Incredible.....
- Huffington Post reports on teachers behaving badly: "Louisiana Teachers Planned Illicit Group Sex With Student: Police". The article states, "Police now say that one of the Louisiana high school teachers accused of having group sex with another teacher and a 16-year-old student had previous sexual encounters with the minor.". There's a video from Huffington Post.