Randomness is everywhere.
But somewhat surprisingly, the mathematical study of probability began relatively recently, when a troubled gambler reached out to a mathematician for help.
[MUSIC PLAYING] Let's start with the word "probability."
It and its relatives "probable" and "probabilistic" derived from the Latin "probabilis," which was primarily used as a morally evaluative term to mean that something was worthy of approval, as in, a probable doctor is a trustworthy one.
More recently, the derivatives of "probabilis" can express a tentative judgment, something more like an intuition or opinion, as in, that hairstyle will probably go out of fashion soon.
And of course, to say something is "probable" can be a statement about factual knowledge, which requires evidence or analytic reasoning.
Throughout much of human history, people consciously and intentionally produced randomness.
They frequently used dice or die-shaped animal bones and other random objects to gamble for entertainment, to predict the future, and to communicate with deities.
Despite all this engagement with controlled random processes, people didn't really think of probability in mathematical terms prior to 1600.
All of the ingredients were there.
People had rigorous theories of geometry and algebra, and the ability to rig a game of dice would have certainly provided an incentive to study probability.
But there's very little evidence that they thought about randomness in mathematical terms.
Suddenly, in the mid-17th century, rigorous probabilistic reasoning began cropping up everywhere.
A correspondence between Blaise Pascal and Pierre de Fermat in response to a gambler's question is often credited as the first formal inquiry into probability theory.
Here's a version of what they wrote about.
Ray and Rusty are playing the most boring gambling game ever.
Each round they flip a coin.
If it's heads, Ray gets a point, and tails, Rusty gets a point.
First person to 100 points wins.
Before the game began, both players contributed $20 to the pot and agreed that the winner would get all $40.
But they get interrupted before either player reaches 100 points.
How should they fairly split the $40?
There are different notions of fairness one can use.
For example, they could split it evenly, or the person with more points can take it all.
But here's a suggestion.
What is each player's expected winnings based on their current scores-- that is, probability of winning?
Check out the link in the description for more information.
Pascal's contributions to probability theory were diverse.
Many of his results, like the correspondence with Fermat, are aleatory, meaning related to chance or dice or luck.
These are likely the kind of calculations you would expect to do in an introductory probability course.
But Pascal thought much more broadly.
The famous Pascal's Wager utilizes a very different notion of "probable"-- one that's epistemic about the certainty or uncertainty of our knowledge.
Pascal was wagering, or betting, on the existence of God.
And he did a calculation of sorts in making this bet.
If he believes in God and God does exist, he reaps infinite rewards.
But if God doesn't exist, it hasn't cost him very much to believe in God.
He doesn't lose much by having a false belief.
Therefore, he concluded, it's rational to wager that God does exist.
There are a lot of different notions of probability, and in the 17th century thinkers like Pascal were working to formalize all of them with varying degrees of success.
Over the next few hundred years, calculations of probability began to accumulate.
But unlike other fields of mathematics, the developing ideas of probability theory didn't all arise from the same foundation.
The definitions and techniques being used were somewhat scattered, or ad hoc.
"The theory of probability as a mathematical discipline can and should be developed from axioms in exactly the same way as geometry and algebra.
This means that after we have defined the elements to be studied and their basic relations, all further exposition must be based exclusively on these axioms, independent of the usual concrete meaning of these elements and their relations."
That quote is from Andrei Kolmogorov's 1933 textbook, "Foundations of the Theory of Probability," which essentially achieved exactly what he's asking for-- a rigorous axiomatic basis for mathematical probability. '
Here's an approximate analogy.
In American English, the word "square" can reference a two-dimensional shape with four corners, a three-dimensional shape with some right angles, any number of approximations to these shapes, a number that can be written as x squared, like 9 or 16, a tool that's used to compute right angles, a solid, hearty meal, or a boring and old-fashioned person.
And that's just the noun form.
But when you're actually studying the properties of a square, none of that's relevant.
The properties of an abstract square are just the consequences of the geometric axioms, basically Euclid's axioms, that govern a square.
It's important that Euclid's axioms are motivated by our intuitive understanding of geometry, and we use that intuition when we're reasoning mathematically.
But it's just as important for the rigor of mathematics to be able to appeal to a strict logical foundation.
How exactly did Kolmogorov axiomatize, or mathematize, probability?
Using a then-recently developed branch of math known as "measure theory."
A measure is a formalized mathematical notion of size.
It takes our intuitions about length, area, and volume and generalizes them.
Probability comes in two flavors.
Discrete questions like how likely is a pair of dice to show 9, and continuous, like questions about a dartboard.
Kolmogorov established a basis for probability theory using measures, which grounds both the discrete and continuous cases.
Let's explore this concept in a very literal case.
You're throwing a dart at this dartboard, and your dart is infinitesimally small, meaning it hits at exactly a point.
For mathematical convenience, we've made the radius of the dart board 1 over the square root of pi, so that the area is 1.
For now, let's pretend you're really bad at darts, and the dart is equally likely to land anywhere on the board.
You're basically throwing blindfolded, and we only count the darts that actually land on the board.
What's the probability you hit this section of the dart board?
1/2.
Why?
The area of the entire dartboard is 1, and the area of this section is 1/2.
The probability is the fraction of the total area.
What's the probability you hit this section?
1/4.
What about this section?
It's still just the fraction of the total area, which in this case is also 1/4.
What's the probability of hitting the point 1/2, 1/2?
0, because the area of a single point is 0.
There's something seemingly paradoxical about this.
It's entirely possible to hit the point 1/2, 1/2, but the probability of doing so is 0.
Saying something has 0 probability does not imply it's impossible.
For every point on the dartboard, the probability that the dart lands there is 0.
But the dart must hit some point.
In fact, even a countable collection of points on the dart board has area 0.
So the probability you hit a point with rational coordinates is 0.
In the previous example, you were throwing wildly at the dartboard, equally likely to hit each point.
That's called a uniform measure.
The area is uniform across the dartboard.
With the uniform measure, the probability associated with each region is the area of the region divided by the area of the entire dartboard.
It's normalizing.
Which is why there is no uniform measure on the entire real number line.
But of course, your throws aren't uniform.
With practice, you have a higher probability of hitting closer to where you aim.
Let's say you always aim for the center.
A quarter of the time your dart lands inside this section, and a quarter of the time your dart lands in this section.
It lands in the outer region the other half of the time.
Instead of thinking about this as different probabilities on the same dartboard, we can modify the way we think about the area of the dartboard.
We use a different measure, our way of viewing area, that more heavily weights the inside of the dartboard.
Make the inside part, where it's more likely to land, bigger.
Now if I throw a dart blindly at this dartboard I have the same probability of hitting each of the colored regions as you do when you throw a dart at the original dartboard.
The more you practice at darts, the more you're literally growing, from a probabilist's perspective, the center of the dartboard.
If we want to mathematically study your darts skills, calculate average throws, the probability of hitting the center ring twice in a row, et cetera, we can just study someone throwing a dart blindly at this dartboard.
That's how the math of probability is laid out.
Probabilities are reflected by larger and smaller areas.
In this example, the points from each dart throw is a random variable.
With some prescribed probabilities, it will return a specific value.
100 if it hits the center, 50 for the next ring, and 0 on the outside ring.
Random variables are coarse-grained.
There's a limited amount of information they can give you.
In this case, it's the value of the ring it lands in.
The random variable doesn't tell you exactly where it is in the ring, just the value.
Kolmogorov realized that you can do this with everything.
Every purely mathematical question about probability can be phrased in terms of a question about sizes-- that is, about measure.
The modern paradigm of mathematical probability is rooted in measure theory.
That's the axiomatic structure from which the theory of probability answers questions.
There are many different intellectual inquiries that appeal to the probabilistic, like whether it's rational to bet on God's existence, how many other intelligent beings exist in the universe, and whether a defendant is guilty or innocent.
These questions can and should be approached scientifically, philosophically, and rationally.
Many of these questions can be formulated and discussed mathematically using the ideas grounded in measure theory.
Part of what I personally love about probability is the connections between the mathematical ideas of probability theory and our scientific and philosophic efforts to understand a world filled with randomness and unknowns.
What's your favorite aspect of probability theory?
Let us know in the comments.
See you next time.
purchase.
Thanks for all the great comments about our episode on structural balance.
Ja-Shwa brought up some really good points that I wanted to share with you guys.
So the first point was that this can happen on a graph with any number of people.
We used six people a lot, and they broke into two groups of three.
But that doesn't have to be the case.
The two groups can be different sizes.
They can be five and one.
You could also do this on a graph with 10 people or 100 people.
That's all fine.
Another point that they brought up was that often in real life, you have a sort of ambivalence towards somebody else.
You're neither friends nor enemies.
And so maybe it would make more sense to have three types of edges-- friends, enemies, and neither.
And that's true.
That's a good point.
That would model real life better.
But it would also make the graph more complicated.
It would make the mathematical model more complicated.
We can study that, but we would be studying a different model.
And then also on the lines of different models, they pointed out that often feelings aren't reciprocal.
One person can be friends with another who thinks that person is their enemy.
People don't always have the same feeling about each other.
And so that could be modeled with a directed graph, which means that there's an arrow pointing in each direction on the edges.
So someone can be friends with someone else, while that person thinks they're enemies.
So that would just be a different model, a different way to study, and it would have different results.
There were a ton of great responses to our challenge, and a lot of really great proofs of what can happen if you have a weakly structurally balanced complete graph.
A lot of you showed that their graph still has to split into factions, but it can have more than two factions.
So every vertex could be its very own group, or they could all be one group.
Doesn't have to just be two.
So of all the great proofs, we wanted to say that our challenge winner is Zutaca this week.
So congratulations.
And finally, we wanted to ask something of you guys.
You often have really wonderful ideas about mathematics and things that you want to learn more about.
So if there's a topic you're particularly curious about and you would like to hear about on a future episode of "Infinite Series," email us at pbsinfiniteseries@gmail.com with the subject line topic.
All right.
See you next week.