In our previous post about March Madness, we assumed that people randomly selected a bracket by chance; we did not take into account that they might remember what they selected previously and adjust their predictions. Let’s say now people become smart and don’t make the same mistake twice. How does this affect our calculations?
March Madness season is back, and with it are disappointments of a busted bracket. I stumbled upon an article (from the Weather Channel — what?!) about the probabilities of getting a perfect bracket: 1 in 9,223,372,036,854,775,808. But just how small is that? Let’s do some math to find out!
…I’m going to calculate the probability that one of my readers from Berkeley has also finished their finals.
Problem statement: Consider final time slots. of the time slots have passed. What is the probability that a student in Berkeley has finished all his/her finals given that he/she has finals?
Assumptions: There are about an equivalent amount of people taking finals in each time slot. There is no systematic “bias” in what final time slot one any student has.
Disclaimer: I’m actually really bad at probability, so correct me for errors in this computation.
Solution: Continue reading →
Consider the daily trek I make from my apartment to places on campus. The eternal question beckons: which is faster — busing or walking? Here I will make a mathematical argument for why it could be the latter or former.
Nietzsche once said that humans are the “as-yet-undermined-animal” (Beyond Good and Evil, Part 3). Now think about that for a second. What he implies is that every other being on Earth has its life cut out for it. We can, to a certain degree, predict what a squirrel does every day. Or a tiger. Or a table. They have purposes, so to speak — they have their destiny. We, on the other hand, have a large degree of autonomy; we make our own destinies.. But we use our autonomy in varied ways — some good, some questionable, and some terrible. And so we wonder about each other.
First of all, what is a square-integrable function? Wikipedia provides this definition:
In mathematics, a square-integrable function, also called a quadratically integrable function, is a real– or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, if
then ƒ is square integrable on the real line . One may also speak of quadratic integrability over bounded intervals such as [0, 1].
The important part is seeing that if the integral of the square of a [Real-valued] function is finite, then the absolute value of the function must converge to zero at +/- infinity, by common sense.
Now the thing that really got me thinking about this was, there have been a lot of good things that have run their courses in my lifetime thus far. Continue reading →
I had this sorta inceptive idea one day. It starts with the idea of the free market juggernaut. It’s an enormously wonderful idea. It’s beyond successful, having laid the foundations for our industrial, scientific, and technological luxuries today. And yet, there’s a fly in the ointment. The more prosperous an economy gets as a whole, the more individual people get screwed. The bigger the gap in poor and wealthy; the more companies can get away with screwing people with dangerous chemicals, outsourcing of jobs, and pollution of our environment. You see, there has to regulation of the economy, because in the end, economies are people…. not numbers. If people get screwed, the economy takes a hit, which is not something numbers alone can tell us. If there is enough gap between rich and poor, there will be civil unrest. As far as I know, no free market model explicitly takes that into account. Economic theories forget about right and wrong, and focus on the numbers.
So that’s the thing right there. People vs. numbers. The idea of right and wrong. People have the understanding of right and wrong. Numbers don’t.
1 / 6+ billion is essentially zero in the number world. It is everything to a person who just lost his/her loved one.
Numbers are cold, hard facts. But the idea of right and wrong — it transcends numbers. Doing the right thing may cost you. That’s numbers right there. But it’s the right thing to do. There seems to be a different logic at work here. The foundation of science is rooted in mathematical logic. The foundation of morality, this idea of right and wrong, is quite different. And the funny thing is, in this whole entire universe, throughout all of space-time, only us, who have been on this tiny blue dot for a blink of an eye… only we possess this different logic. At the same time, only we possess the number logic too. So who’s to say one logic is more valid than another?
I hate knocking on mathematics here. I love the intrinsic beauty in mathematics, and its broad applications in describing the world. But it can’t touch morality. Morality is just one of those really weird things. It’s there, and we all know it’s there, and it makes a difference, as we saw in the free market. But all that we have worshiped in the past couple of centuries — the scientific method, the grounding of our understanding in mathematical logic — it doesn’t lay a hand on morality. The *moral* of the story: numbers do not span the entire space of reality. If we can build up a theory of everything with numbers, can we build up a theory of everything with morality? Or do we need both? Does one simply have more applicability in one domain, whereas the other in another domain? Questions..
Maybe math is the tool to understand the outside, the universe… whereas morality is the tool to understand ourselves. I dunno.
You see, this is important, because if math doesn’t explain ALL of the universe, you’re missing out if you’re saying science is the end-all-be-all. Religion has been universal because humans couldn’t explain what was outside. It still is universal because, while we’ve extensively used mathematical logic to understand what’s outside of us, we haven’t used moral logic to understand what we’re made of.
And in the end, both logics only exist on one tiny blue dot in the middle of a vast universe. We don’t know where either logic comes from.
Food for thought.