Probability - Episode 15 - Maths

Imagine a massive pile of gold coins sitting on a rough wooden table. We are in a dimly lit 17th century French parlor. >> Oh, that sets the scene perfectly. >> Right. The dice are rolling. Fortunes are just changing hands. By the minute, the tension in the room is totally palpable. >> And then suddenly everything just stops. >> Exactly. Before the final roll can even be made, the game is abruptly stopped and the players are just staring at this massive pot of gold >> wondering who gets the money. >> Yeah. Because you can't just give it to the person who happens to be in the lead, right? >> Because the other player still had a mathematical chance of a comeback, >> right? But you also can't just split it down the middle. The player in the lead had a vastly superior position. >> Exactly. You have to somehow measure the exact value of a future that well never actually happened, >> which is an incredible puzzle. >> It really is. And figuring out how to divide that interrupted pot of gold didn't just settle some random casino dispute. It literally birthed the mathematical foundation of the trillion dollar global insurance industry. >> It's just one of the most profound paradigm shifts in human history. Honestly, we're looking at the exact moment when mathematics stepped out of the pristine, completely predictable laboratory of classical physics. >> Right. The world of perfect certainty. >> Exactly. And it stepped into the chaotic, messy, entirely unpredictable real world. >> And that is exactly what we are unpacking today. Welcome to this deep dive into the source material. >> It's going to be a fun one. >> It really is. Today's mission is to uncover the absolute bedrock of the actuarial profession. We're relying on an incredible compilation of research and historical analysis centered around the concepts explored in the mathematics of uncertainty. >> Especially for any actuarial students listening today, this is the core of everything you'll do. >> Absolutely. We are going to explore how humans finally learn to measure the unknown, turning chaos into exact, actionable fractions. I want you listening right now to think about your own life. >> We all deal with it constantly. >> We do. You navigate risk every single time you merge onto a highway. Um, every time you sign a mortgage, every time you start a business, you carry this immense burden of the unknown. >> And today, we're going to break down the exact mechanical laws of chance that actuaries use to mathematically shoulder that burden for us, >> allowing modern society to function without just, you know, paralyzing fear. >> Right? And to truly understand how actuaries engineer that financial security today, we have to look at the origins of these tools, >> which is wild, by the way. >> It is because what's fascinating here is that this specific branch of mathematics was not developed by academics sitting in ivory towers theorizing about the cosmos. >> Not at all. >> It was developed by people who had real skin in the game. It was forged in the gambling dens of Renaissance Europe, >> which is just the best origin story for a highly rigorous scientific discipline. I mean to understand the sophisticated algorithms running massive global risk models today we actually start with a 16th century Italian physician >> Jerommo Cardano. >> Yes Cardano. Now our source material highlights that Cardano was a profoundly gifted analytical mind. I mean he's the same mathematician who famously wrestled with the concept of imaginary numbers. >> A literal genius of his time. >> But he had a fatal flaw that just consumed his life. >> Yeah. a debilitating gambling addiction. Cardano was a man torn between his immense intellect and his absolute compulsion for the dice tables. >> It's such a tragic but relatable human conflict. >> Truly, but his analytical mind simply could not accept losing blindly. At that time, you know, the prevailing human belief about chance was entirely mystical. >> Like it was literally up to fate. >> Exactly. If you rolled a double six, it wasn't seen as a statistical inevitability. It meant the gods favored you on that particular Tuesday. >> Chance was just viewed as the unpredictable whim of fate, >> right? It was just luck. You couldn't predict it. You couldn't control it. You just prayed for it. But Cardano does something that is so simple in hindsight, yet completely revolutionary for the 1500s. >> I love this part. What does he do? >> He decides to systematically write down every single possible combination of a dice roll. >> He essentially forces the chaos of the gambling table to reveal its underlying architecture. >> Exactly. By meticulously recording these combinations, Cardano experiences a massive paradigm shift. He realizes that chance is not the whim of the gods. >> It has a hidden highly mechanical structure. Yes, let's really visualize what he did. Um because it demystifies the whole concept. If you take two standard six-sided dice, you might think, well, the numbers go from 2 to 12. It's just random what I get, >> which is how everyone else at the table was thinking, >> right? But Cardano mapped out the grid. He saw that there are exactly 36 possible ways those two dice can land >> precisely. If you want to roll a two, there is only one specific reality where that happens. The first dice must show a one and the second die must show a one. >> So that's one outcome out of 36. >> Right. But if you want to roll a seven, well, the mechanical structure of the dice offers you many more pathways. >> Exactly. You could roll a one and a six, a six and a one, a two and a five, a five and a two, a three and a four, or a four and a three. >> You're just walking down the grid. Yeah, there are six distinct combinations out of the 36 that result in a seven. So, the chance of rolling a seven is 6 out of 36 or, you know, one in six. >> And right there, Cardono realizes the game isn't random. The dice are heavily biased toward the middle numbers by the sheer laws of physical combinations. >> He saw the gears turning beneath the surface of the game. >> He really did. And that realization is the very spark of probability. Chance transition from a mystical uncontrollable force to a mechanical decodable entity >> which gives you immense power. >> Oh, absolute power. Once you know that a seven will appear on average six times more often than a two, you know exactly which wages to accept and which to decline. You have a mathematical edge. >> And he actually wrote a manuscript detailing all of this, right? The very first text on the mathematics of chance. >> He did write it, but he never published it. >> Wait, really? He just kept it. He kept the whole thing a complete secret in his coat pocket until long after he died >> because understanding the geometry of chance in an uncertain world is an incredibly lucrative superpower. >> Exactly. Cardano wanted to keep that edge exclusively for himself to fund his gambling. It perfectly illustrates the immediate raw practical value of this knowledge. >> That's wild. But the true official birth of probability like where it finally entered the broader intellectual discourse and changed the world happened about a century later. Right. >> Yes. Which brings us back to our parlor in 17th century France. >> Ah right. Back to the pile of gold on the table. The year is 1654. >> Right. And the man staring at the gold is the Shiovali Deere, a French nobleman, a writer, and a notoriously heavy gambler. >> Of course, he's a gambler. He is the one who experienced this famous interrupted game we talked about in the start. >> Yes. The problem of points as it became known in mathematical history. The Shioval de was playing a highstakes game of chance. >> And what were the rules? >> Well, they were playing a series of rounds and the first person to reach a certain number of points would take the entire pot of money. But for whatever reason, the game was forced to stop before either player reached the winning score. Okay, so let's put some specific numbers to this to really see the dilemma, especially for our actuarial students trying to grasp the foundational logic here. >> Let's do it. >> Let's say they are playing a simple coin flipping game. First person to win three flips gets a 100 gold pieces. Player A has won two flips. Player B has won one flip. Bam. The game is interrupted. >> The chalier is looking at the 100 gold pieces. How do you divide it fairly? It is the ultimate puzzle of unrealized potential. Player Agues, well, I was winning. I need just one more point. I should get the vast majority of the money. >> But player B argues, wait, I still had a chance. If the game continued and I won the next two flips, I would take the whole pot. You can't just shut me out entirely. >> It's a valid point. The Chavevelier, recognizing that this was a deeply complex, logical trap, took the problem to one of the greatest minds of the era, >> Bla1 Pascal. Bla1 Pascal, a towering figure in mathematics and philosophy. And Pascal becomes entirely obsessed with this interrupted game. >> Totally obsessed. But even Pascal realizes he needs a sounding board. So he begins writing letters to another genius, Pierre de Ferat. >> What followed was an extraordinary series of letters bouncing between these two brilliant minds. >> Right through this correspondence, they weren't just trying to figure out who gets the gold. They were actively inventing the mathematical framework required to evaluate the future. >> Here's where it gets really interesting. Let's actually walk through the math they developed to solve this because it is so incredibly elegant. >> I love this part. >> Going back to our example. Player A has two points. Player B has one point. They need three to win. Pascal and Fermat realized you don't look at the past, right? >> No, you have to map out the hypothetical futures that would have happened if the game continued. >> You map the branching tree of possibility. So, what happens on the very next hypothetical flip? >> Well, the next flip is either heads or tails. Let's say player A gets a point on heads. If it's heads, player A gets their third point. Game over. >> Player A wins the 100 gold pieces in that specific future universe. >> Yes. But what if that next hypothetical flip is Tails? Then player B gets a point. The score is now tied two to two. The game still isn't over, >> right? So, we had to go one more flip into the future. Yeah. From that tied two to2 state, the next flip decides absolutely everything. >> Exactly. If it's heads, player A wins. If it's tails, player B wins. >> So, if we look at the entire landscape of possible futures from the moment the game was interrupted, there were essentially four equally likely paths the game could have taken. >> Let's count them out to be precise. >> Okay, let's do it flip by flip. The next two flips have four combinations. Heads heads, heads, tails, tails, heads, and tails tails. Right? >> If it's heads, heads, A wins. If it's heads, tails, A wins on the first flip anyway. If it's tails heads, B ties it, then A wins. And if it's tail tails, B ties it, then B wins. >> Perfect. So out of those four exact future scenarios, player A walks away with the prize in three of them. >> And player B only walks away with the prize in one of them. >> Therefore, the exact mathematically fair division of the 100 gold pieces is not 50/50, and it isn't based on their past scores. >> It is based precisely on their ownership of the future. >> Player A owns threearters of the future probability. Player B owns one quarter. Player A gets 75 gold pieces. Player B gets 25. >> That is just brilliant. They solved it. But stepping back for a second, I really want to emphasize the sheer scale of what happened here. >> It's massive. Are we really saying that the entire foundational theory of probability like the exact same rigorous mathematics that today runs trillion dollar global insurance markets the math that decides our healthcare premiums was essentially invented via mail correspondence just to settle a casino dispute for a French nobleman. >> That is precisely what happened. It is a stunning historical reality. >> Wow. >> Through those letters, Pascal and Ferment did something revolutionary. They quantified hope. that quantified risk. >> Before 1654, the future was viewed as just an indivisible, impenetrable block of mystery. You either win or you lose eventually. >> But Pascal and Format proved that the future can be sliced up. It can be divided into exact measurable fractions. They proved you can weigh a reality that has not even occurred yet. >> It is a massive conceptual leap. They took the unknown and gave it a geometry. But as our source material points out, there is still a very wide gap between two French mathematicians exchanging letters about a parlor game in the 17th century and the rigorous, highly structured, airtight science that actuaries use today. >> Yes, Pascal and Ferat gave us the intuition and the initial methods, but their work was essentially a collection of clever solutions for specific games. >> Right. It wasn't a unified theory yet. >> No, it wasn't. To bridge the gap from gamblers's intuition to a perfect unified science, we have to move forward in time to the 1930s. And we have to look to a Russian mathematician named Andre Kaggarov. >> Andre Kogorov for anyone listening right now studying risk statistics or actuarial science. This is where the modern era officially begins. >> Without a doubt, Kogarov looked at all this fragmented math of chance and realized it needed a bedrock foundation. It needed absolute laws. He recognized that probability was floating in a conceptual void. It lacked the rigorous definitions that guided geometry or algebra. >> So in the 1930s, Klemarov published the axioms of probability. These are the unbreakable rules, the absolute laws of physics for the universe of chance. >> The axioms of probability. Let's really break down how Kaggorov built this because the way our source material visualizes it makes these incredibly dense mathematical concepts so accessible. It starts by defining the core arena where chance happens. The sample space. >> The sample space. >> Yes. And for any actuarial student, the sample space is the foundation of every single probability calculation. If you do not define your sample space correctly, every subsequent calculation is completely meaningless. >> To visualize this, I want you to imagine a massive empty room. You're standing at the threshold of this gigantic, brightly lit, entirely empty room. >> I like this analogy. In Kgra's mathematics, this room represents the absolute totality of every single thing that could possibly happen in the specific scenario you are analyzing. It is the container for all potential reality. >> Okay, let's populate that room. If you are rolling a standard six-sided die, the moment that die leaves your hand, the empty room instantly populates with exactly six potential futures. >> You can see them standing there. A future where the dieice shows a one. A future where it shows a two, a three, a four, a five, and a six. >> And this is the crucial part. Nothing else is in that room. >> A future where you roll a seven does not exist in this specific room. >> Right? A future where the die shatters into dust does not exist either. The room contains only those six mutually exclusive exhaustive outcomes. That is your sample space. >> Now, let's elevate that from a casino game to the reality of the actuarial profession. >> This is where it gets real. If an insurance company writes a policy for a new driver, the room looks completely different. The sample space for that one-year policy contains a vast array of futures. >> It contains a future where the driver has zero accidents. It contains a future where they rear end someone at a stoplight. It contains a future where the car is totaled in a hail stom. >> The entire universe of possibility for that specific risk over that specific time frame is locked inside that room. >> Exactly. Okay. So, we have our room. we know exactly what can happen. The next step KGR formalized is the concept of an event. >> Now in normal conversation, an event is just a thing that happens like a birthday party or a concert. But in the axioms of probability, an event has a very strict mathematical definition. >> In this framework, an event is simply the act of selecting a specific subset of the futures inside your room. >> I picture it like walking into that massive room with a bright red marker. >> Go on. >> Let's go back to the die roll. The room has six futures standing in it. I want to analyze the event of rolling an even number. So I walk up to the future where the dieice shows a two and I draw a big red circle around it. I draw a circle around the four. I draw a circle around the six. I have grouped them together. That circle I just drew is the mathematical definition of an event. >> Precisely. You have defined a subset of the sample space. And this brings us to the ultimate definition. What is probability itself? How does Karov assign a definitive number to that red circle you just drew? >> He defines probability not as a feeling or a likelihood, but strictly as a set function. >> A set function? It sounds incredibly intimidating. >> It really does. But let's demystify it. Think of a set function as a machine. It's a very simple, very precise machine sitting just outside the door of our massive room. >> This machine has one singular purpose. It looks at the red circle you drew your event and it mathematically evaluates the outcomes inside that circle relative to the total number of outcomes in the entire room >> and then it assigns a weight to your circle. >> Right? It calculates a fraction. The numerator is the number of futures inside your circle. The denominator is the total number of futures in the room and it spits out a percentage strictly between 0 and 100. >> And Kongarov's axioms state that these boundaries are absolute. [snorts] If the machine assigns a zero, it means that event contains no futures. It is mathematically impossible. >> And if the machine assigns a 100% or one, it means your circle encompasses every single future in the room. It is absolute certainty. >> So for our even number circle, the machine sees three futures inside the red circle. It sees six total futures in the room. 3 / 6. >> The machine spits out 50%. Boom. >> We've mathematically quantified the unknown. This simple mechanism, this set function evaluating the geometry of the room is the core engine of all actuarial science. But the real world is rarely as simple as drawing one single circle. >> Right? Which leads us to the addition rule. Because if I'm an actuary, I might need to know the probability of a client experiencing event A or event B. And the intuitive but often extremely dangerous approach is to simply run the machine for event A, run the machine for event B, and add the two percentages together. >> Okay, let me test my understanding here. >> Let's say we are looking at a standard deck of 52 cards. My massive room has 52 futures in it. >> Got it? >> I want to know the probability of drawing a heart or drawing a king. So, I take my marker. I draw a circle around the 13 hearts. The machine says that's 13 out of 52, which is 25%. >> Okay. >> Then I draw a second circle around the four kings in the deck. The machine says that's four out of 52. About 7.7%. I just add them up. 13 + 4 is 17. 17 out of 52. Done. Easy. >> Hold on. Let's look very closely at the geometry of those circles you just drew inside the room. You drew a circle around the 13 hearts. You drew a circle around the four kings. Where do those two circles intersect? >> Oh, wow. What is inside the overlapping space of that vin diagram? >> Ah, the king of hearts. >> Exactly. When you drew the first circle for the hearts, the king of hearts was inside it. Machine counted it. When you drew the second circle for the kings, the king of hearts was inside it again. The machine counted it a second time. >> Oh wow. I double counted a future. I literally created a ghost future that doesn't exist, mathematically inflating my probability. >> And this is a fatal error in risk assessment. If you blindly add probabilities without accounting for the intersection of the events, your entire actuarial structure collapses. >> So the true addition rule requires you to add the probability of event A, add the probability of event B, and then subtract the probability of the intersection where event A and event B occur simultaneously. >> So the correct math is 13 hearts plus four kings minus the one king of hearts that got double counted. 16 out of 52, not 17. It is such a subtle distinction, but when you are pricing billions of dollars of whisk, that double counting error would result in catastrophic financial losses. >> It ensures that the foundation of your mathematical model perfectly mirrors the physical reality of the sample space. You must respect the boundaries of the room. >> Okay, let's unpack this. We've established the rules of the room. We know how to draw circles. We know how to run the machine. We know how to add things up without double counting, right? >> But this is all happening in a static room. The 52 cards are just sitting there. What happens when reality starts unfolding? What happens when we get a piece of new information that completely changes the environment? This is where we move from static probability into the most dynamic powerful concept in our source material. This is where we introduce conditional probability and fundamentally the mechanics of how the universe shrinks. >> I want to use a slightly different analogy to explain this because I think it helps visualize the mechanics better. Instead of just a shrinking universe, let's think of conditional probability as a mathematical filter or a civ. >> A filter is a highly accurate way to describe it. >> Let's go back to our deck of 52 cards. The room has 52 possible futures. We want to know the probability of drawing a king. As we established, the baseline probability is four out of 52, >> right? The machine spits out about 7.7%. >> But now we introduce a condition. Let's say I draw a card from the deck. I look at it and I keep it hidden from you. But I give you a massive hint. I tell you the card I am holding is a face card. >> The moment you introduce that new verified evidence, the reality of the room is violently altered. The filter has been applied. >> Right? The condition it is a face card acts as an absolute mathematical civ. It immediately filters out and deletes every single future that does not meet that condition. >> The future where the card is a two of clubs gone. future where it's a seven of spades. Deleted the 10 of diamonds vanished. >> You are no longer standing in a room with 52 futures. Your denominator has fundamentally changed. The new evidence has forcefully ejected 40 cards from the sample space. >> The new room only contains 12 futures. The jack, queen, and king in all four suits. >> And this changes everything about our calculation. We are still looking for the event of a king. The number of kings in the deck hasn't magically changed. There are still exactly four kings, >> but the environment they exist in has shrunk. >> Exactly. Our machine now looks at the four kings, but it divides them by the new denominator of 12. >> Your probability of holding a king jumps from 4 out of 52, which is less than 8%, to 4 out of 12, >> which is 33.3%. >> The probability literally quadrupled, not because the target changed, but because the universe of alternatives was filtered away by new evidence. This mechanical process of updating a probability based on new conditions is the direct gateway to what is universally considered the most powerful tool in modern statistics. >> Baze theorem. >> Yes. Named after Thomas Ba, an 18th century English minister, >> which is just another incredible cast member in our story. >> Mhm. We have the gambling physician Cardano, the nobleman Marie, the rigorous Russian Kulmogarov, and now an English minister from the 1700s providing the ultimate skeleton key for data science. >> It's quite a lineup. >> I love the phrasing used to describe Bae theorem in the research. It calls BA theorem the mathematics of changing your mind. >> It is the perfect description. Human psychology is notoriously flawed when it comes to changing our minds. We suffer from confirmation bias. We cling stubbornly to our initial beliefs or we overreact wildly to a single piece of scary news. >> BA theorem removes the human emotion entirely. It provides a flawless logical framework for exactly how much you should change your mind when you encounter new evidence. >> Let's really break down the anatomy of bees theorem because it introduces some terminology that is absolutely vital for anyone navigating risk. There are three main components. >> The prior, the likelihood, and the posterior. The mechanism is elegant. It starts with the prior probability. This is your initial belief. It is the baseline probability of an event happening before you see any near evidence. >> Then you observe something in the real world. You get new data. You have to evaluate the likelihood of seeing that exact data. if your initial belief were true versus if your initial belief were false. >> And BA's theorem is the mathematical engine that grinds the prior and the likelihood together, filtering the sample space to produce the posterior probability. >> The posterior is your perfectly updated new belief. >> To really cement this, we have to look at a classic statistical trap that Baze theorem solves. It's called the false positive paradox or the base rate fallacy. This is the moment where Beijian math usually causes a massive aha realization for people. Let's walk through this carefully. Imagine a scenario involving a rare disease. >> This is a perfect application. Let's establish the prior probability. >> Okay, let's say medical data shows that exactly 1% of the population actually has this specific rare disease. That is our prior. If I pick a random person off the street, there is a 1% chance they are sick. >> Okay. >> Now, let's introduce a medical test for this disease. This test is highly accurate. It is 99% accurate. >> Let's define what that 99% accuracy actually means mechanically. If you have the disease, the test will correctly identify it 99% of the time. >> Right? >> If you do not have the disease, the test will correctly tell you that you are healthy 99% of the time. It only makes a mistake, a false positive or a false negative 1% of the time. So here is the scenario for you the listener. You go to the doctor. You take this 99% accurate test. The doctor comes back into the room looking grim and says the test came back positive. >> What is the actual mathematical probability that you have the disease? >> Human intuition immediately screams, well the test is 99% accurate. It's positive. So I have a 99% chance of being sick. >> That is the intuitive emotional reaction. But it is entirely mathematically wrong. It completely ignores the structure of the room. It ignores the prior. >> Let's run this exact scenario through the Beijian filter and look at the absolute numbers. >> Let's use a population of 10,000 people to make the math visible. We have 10,000 people in our massive room. >> Step one, apply the prior probability. We know 1% of the population actually has the disease. What is 1% of 10,000? >> 100. So, inside our room of 10,000, there are exactly 100 truly sick people and 9,900 perfectly healthy people. >> Now, step two, we administer this 99% accurate test to everyone in the room. Let's look at the 100 truly sick people first. >> The test is 99% accurate. So, out of those 100 sick people, the test will correctly flag 99 of them with a positive result. >> Okay, we have 99 positive tests sitting on the table. But now we have to test the healthy people. There are 9,900 perfectly healthy people. >> And the test is 99% accurate for them, too. Which means it makes a mistake, a false positive 1% of the time. >> What is 1% of 9,900? It is 99. >> Precisely. The test will accidentally generate 99 false positive results from the massive pool of healthy people. >> Okay, let's pause and look at what is sitting on the doctor's desk. The doctor has a stack of positive tests. How many are in the stack? >> There are the 99 true positives from the sick group plus the 99 false positives from the healthy group. There are 198 total positive tests. >> And you are holding one of those positive tests, >> right? I know I am somewhere in that stack of 198 positive results. >> But what are the chances I am one of the truly sick people? The math is exactly 99 true positives divided by the 198 total positive >> which is exactly 50%. >> 50%. Your probability of actually having the disease isn't 99%. It's a coin toss. It's 50/50 >> because the sheer volume of the healthy population, the base rate, the prior overwhelms the accuracy of the test. The 1% error rate applied to 9,900 people generates just as many positive results as the 99% accuracy rate applied to the tiny group of 100 sick people. >> That is staggering. Without B theorem, a doctor might panic and prescribe drastic treatments based on a 99% intuitive assumption. With B theorem, the doctor mathematically understands that a positive result merely shifts the probability from a 1% baseline up to a 50% concern requiring a second test. >> B theorem forces us to respect the base rate. It forces us to synthesize our historical reality the prior with our new localized evidence to arrive at the correct posterior truth. >> It is an indispensable tool for actuaries who are constantly bombarded with new data that must be weighed against historical trends. which perfectly transitions us to the next massive challenge in evaluating risk. BA's theorem is phenomenal for updating a single belief, a single event. But reality isn't a single isolated event. >> No, the real world is a sprawling web of countless events happening simultaneously. How do these different events interact with each other inside the mathematical universe? To map the real world, actuaries must master the distinction between two fundamentally different types of interaction, independence and correlation, >> or as we might visualize it, events that are moving blind versus events that are moving together. >> Moving blind versus moving together. Let's look at the mathematical definition of independence first. The classic, universally understood example is flipping a coin multiple times. >> A standard coin flip, the ultimate independent event. If I flip a coin and it lands on heads and then I pick it up to flip it a second time, KMurov's axioms tell us that the universe of the second flip is completely totally isolated from the universe of the first flip. >> The coin has no memory. The universe of the second flip is completely blind to what just happened. The probability remains exactly 50% for heads and 50% for tails, regardless of whether you just flipped one head or 10 heads in a row. This sounds obvious, but human psychology aggressively fights the mathematics of independence. >> Oh, absolutely. >> If you go into a casino, let's bring Cardano and Marray back into the room, and you stand at a roulette wheel, you watch the ball land on black, then black again, then black a third time, a fourth time, a fifth time, five blacks in a row. >> Every human instinct, every emotional fiber in a gambler's brain screams that red is now due. The brain insists that the universe must balance itself out immediately. It's called the gamblers's fallacy. People will bet their life savings on red because they feel the pressure of the past rolls. But the wheel is an inanimate object. It is moving blind. >> The probability of black on the sixth spin is the exact same as it was on the first spin. Treating independent events as if they are somehow magically connected is a mathematically fatal flaw. >> But the danger works in both directions. Assuming events are connected when they are actually independent will cost you money at the casino. However, assuming events are independent when they are actually deeply connected when they are deducted, correlated will completely bankrupt an insurance company. >> Let's explain correlation. How do events move together? The classic example is agricultural risk. >> Let's say you are tasked with measuring two specific variables in a farming valley. Variable A is the total amount of rainfall the valley receives in the spring. Variable B is the total tonnage of the wheat harvest in the fall. These two variables are not blind to each other. They are intimately tethered to the same underlying physical reality. >> If the valley experiences a season of perfect abundant rainfall, the mathematical probability of a massive weed harvest in the fall skyrocket. >> And conversely, if there is a severe prolonged drought, the probability of a failed harvest approaches certainty. The events move together. They are highly correlated. >> The rainfall is the underlying engine driving the outcome of the harvest. So understanding the absolute distinction between independent events and correlated events is the final piece of the puzzle. We have the history, we have the axioms, we have the Beijian filter, we have the correlation. >> We have built this incredible theoretical architecture. >> Now we have to ask the ultimate question. How does an actuary actually weaponize this mathematics in the real world? How do these theories translate into a literal business model? This is where we transition from the theory of chance to the applied superpower of the actuarial profession. The underlying truth is that without this specific mathematical framework, the concept of modern insurance literally does not exist. >> It is the difference between gambling and engineering. Actuaries are the architects of financial security. Let's look at the most fundamental application of this power. Pricing insurance. How do you decide how much a human being should pay for a policy? The foundational rule of insurance pricing is absolute. You cannot sustainably charge a premium if you do not accurately know the expected cost of the claim. >> And how do you calculate that? By using Kaggarov's set function machine to monetize the future. The core formula is straightforward but profound. The expected cost equals the severity of the potential loss multiplied by the probability that the loss will actually occur. >> Severity multiplied by probability. Let's apply this to a life insurance policy which provides a very stark clear example of the mathematics. >> Okay. Let's say an individual wants to purchase a one-year life insurance policy with a payout or severity of $100,000. If the individual passes away during that year, the company must pay out $100,000. >> But the company isn't going to charge the individual $100,000 for the policy. >> No. Because the event of the individual passing away in that specific year is not a certainty. It is a risk bounded by probability. >> This is where the actuary steps in. They consult their mortality tables which are essentially massive data sets fed into the axioms of probability. Let's say the data shows that for a person of this specific age and health profile, the probability of passing away within the next year is precisely 1%. 01. >> So you apply the formula, you multiply the severity $100,000 by the probability 01. 100,000 multiplied by 1% is $1,000. >> That $1,000 is the expected cost. It is the raw mathematical value of that specific slice of the future. It becomes the foundational baseline for the premium they will charge. >> But here is the critical operational question that eliminates the genius of the actuarial model. If the company charges that individual $1,000 and the individual actually does pass away, the company has to pay out $100,000. >> They just lost $99,000 on that single transaction. >> Right? So, how is that a viable stable business model? How do they survive that variance? >> This is where we have to introduce the ultimate magic trick of the actuarial profession. The law of large numbers. This is the why and the how behind the entire industry. The law of large numbers is the bridge between chaotic individual unpredictability and perfect macrolevel certainty. >> If the insurance company only sells one life insurance policy, they're essentially just gambling. They're playing roulette. They collect $1,000 and pray they don't have to pay out a h 100,000. >> The variance, the potential swing between profit and catastrophic loss is massive. >> But actuaries do not build businesses on single policies. They build businesses on volume. What happens if the company sells that exact same policy with the exact same 1% probability to 100,000 different independent people? >> Let's do the math on the aggregate. They collect $1,000 from 100,000 people. That is $100 million in premium revenue collected upfront. >> Now, what is the expected payout? We have 100,000 people. The probability of death is 1%. >> 1% of 100,000 people is 1,000 expected deaths. and the payout for each death is $100,000. >> 1,000 deaths multiplied by $100,000 is $100 million in expected payouts. The revenue perfectly matches the expected cost on a macro scale. >> And this is exactly what the law of large numbers dictates. As the number of independent trials, or in this case, the number of independent policy holders increases, the actual observed results will converge perfectly upon the mathematically expected results. >> The variance shrinks to almost nothing. The chaos of individual human life is smoothed out into a highly predictable, manageable statistical curve. >> The larger the pool, the sharper the precision. They can predict with astonishing accuracy that exactly 1,000 people out of the 100,000 will pass away. They don't know which 10,000 people, and they don't need to. The math protects the entire pool. It is absolute genius. >> It is the democratization of risk. But the pricing model isn't always static. It has to adapt to human behavior over time. And this brings us to the second major application which takes us right back to our 18th century minister professional practice and Beijian credibility. >> Yes. How do you dynamically adjust pricing when reality provides new evidence? Let's take auto insurance. You the listener imagine you just got your driver's license. You apply for insurance. The actuary has no idea who you are. You have zero personal driving history. >> In the absence of individual data, the actuary must rely on the prior. They assign you the average base rate for all new drivers. They assume you represent the historical average probability of causing an accident. >> But then you actually start driving. Reality happens. Let's say over the next 3 years you get into two atfall car accidents. >> So the universe has just provided new highly specific evidence about your individual risk profile. >> The actuary now faces a dilemma. They can't keep charging you the average rate because you have empirically proven to be riskier than average. But they also shouldn't throw out the historical data entirely and assume you're going to crash twice every 3 years forever because maybe it was just an unusual string of bad luck. >> This is where actuaries apply Beijian credibility. It is the literal application of BA's theorem to insurance premiums. The actuary takes the prior the industry average and they blend it with the likelihood of your new evidence. The two crashes. >> They run your file through the mathematics of changing your mind. >> Precisely. And the Beijian formula tells them mathematically, emotionlessly exactly how much weight to give your personal history versus the industry average. It outputs a perfectly updated posterior probability. >> It shrinks the universe around your specific behavior, resulting in a new personalized premium that accurately reflects your mathematically updated reality. >> It's a self-correcting machine that constantly fine-tunes the pricing of risk. It is elegant, but the scope of the actuar's job scales up far beyond individual car crashes. And this brings us to the final and perhaps most critical application of the math we've discussed today. Enterprise risk management or ERM. >> ERM is where we see the absolute existential importance of understanding independence versus correlation. This is where actuaries protect the survival of the insurance company itself. >> To understand erm, we have to look at how companies aggregate risk. Let's imagine an insurance company writing 1,000 homeowner policies. In scenario A, the actuary ensures these thousand policies are spread out evenly across the globe. You have a house in Tokyo, a house in London, a house in Toronto, a house in Sydney. >> In this scenario, what is the mathematical relationship between those risks? >> They are completely independent. They are moving blind. If a kitchen fire breaks out in the house in Tokyo and burns it to the ground, that fire does not magically cross the ocean and increase the probability of a fire in the London house. Because the risks are independent, the law of large numbers functions perfectly. The variance is controlled. The company can reliably predict how many isolated fires will happen globally in a year and hold just enough capital reserves to pay those specific claims. >> But then we look at scenario B, the nightmare scenario. What if the company writes 10,000 homeowner policies, but instead of spreading them globally, they write all 10,000 policies on the exact same stretch of coastline in South Florida? The mathematical environment has fundamentally changed. We are no longer dealing with independent coin flips. We are dealing with the rain and the wheat harvest. >> These risks are highly correlated. They are tethered to the exact same underlying threat, the Atlantic hurricane season. >> If a category 5 hurricane makes landfall on that specific coastline, it does not destroy one isolated house. It destroys all 1,000 houses simultaneously. If the actuaries mistakenly priced those Florida policies, assuming they were independent, assuming they could rely on the law of large numbers to smooth out the losses, they would be walking the company into an absolute bloodbath. >> To visualize the financial mechanics of this, think of an insurance company's capital reserves like the vault in a bank. In normal independent conditions, a few people come to the bank every day to withdraw money. These are your isolated house fires. The vault always has enough cash to handle the daily predictable flow. >> But a correlated catastrophe, a hurricane hitting a thousand houses at once, is the equivalent of a massive bank run. Every single policy holder lines up at the vault on the exact same day, demanding their massive payout simultaneously. >> The liquidity instantly dries up. The capital reserves are vaporized. The company becomes insolvent. This is not just a theoretical exercise. The failure to properly model correlated geographic risk has destroyed real companies. >> We have to talk about Hurricane Andrew in 1992. It is the textbook case study for correlated risk failure. Before Andrew, many insurance companies had massive concentrations of policies in South Florida. Their model simply did not adequately account for the extreme correlation of a storm of that magnitude. >> When Hurricane Andrew hit, it caused over $15 billion in insured losses at the time. It triggered a literal wave of bankruptcies. At least 11 insurance companies went completely insolvent because they had essentially hoarded interconnected risks. Their bank faults were emptied in a matter of hours. >> It is the ultimate lesson in the power of correlation. After Hurricane Andrew, the actuarial profession revolutionized enterprise risk management. They utilized advanced geographic correlation mapping. They leaned heavily into reinsurance, which is essentially insurance companies buying insurance for themselves from massive global pools to offload concentrated risks. They realize that you cannot play games with correlated probabilities. >> Actuators are stationed at the very center of the global economy, constantly running these diagnostics. They are measuring the dimensions of the empty rooms, checking the boundaries of the set functions, updating the Beijian prior with new climate data, and frantically stress testing for hidden correlations. All to ensure that the mathematical safety nets of society actually hold firm when the worst happens. >> It is an immense almost staggering responsibility. And it really brings us to the ultimate message of our deep dive today. When you look at the journey from Cardano to Kaggorov to modern risk management, you realize that probability is not just the mathematics of gambling anymore. >> It was born in the shadows of vice. Certainly, it started as a tool to gain an edge at the dice tables, but its ultimate destination has been the preservation of society. The mathematics of chance has evolved into the mathematics of hope, of profound preparation, and of societal resilience. >> The mathematics of hope. I want to speak directly to you, the listener, as we wrap this up. Think about the freedom you have in your life. The reason you can buy a house without living in paralyzing terror of a fire destroying your family's entirely net worth. >> The reason you can open a small business without the fear of a single lawsuit leaving you destitute is because an actuary has mathematically quantified that risk. They have run the numbers, apply the axioms, leveraged the law of large numbers and built a financial product to shoulder that terrifying unknown for you. They have taken the chaotic, terrifying future and they have successfully divided it into exact, manageable, measurable fractions. >> They took the hidden mechanical gears of chance that Cardano first glimpsed in the 16th century and they use those gears to build the most robust engine of financial stability the world has ever known. It is an extraordinary testament to the power of human intellect to tame the unknown. >> It really is. It takes these cold, rigorous equations and translates them into genuine human security. >> But you know, as we close the book on this steam type, I can't help but be left with one final slightly provocative thought to mull over. We spent a lot of time talking about BA theorem today. We talked about how it is the perfect mechanism for changing your mind based on taking historical averages, the prior, and updating them with new data. >> Yes. the constant logical refinement of belief based on history and observation. >> Right? But here's the question that keeps me up. If the entire actuarial profession relies so heavily on historical averages to build their initial models, what happens when the underlying physical conditions of the world begin changing faster than the historical averages can possibly keep up with? >> You are referring to the shifting baseline. >> Exactly. Think about the Florida coastline from our correlated risk example. What happens to the Beijian math when the global climate affecting those oceans shifts so rapidly that the 100red-year history of hurricanes no longer accurately predicts the next 10 years of hurricanes? If your prior probability is fundamentally broken by a changing climate, your perfectly updated product, posterior probability is going to be dangerously wrong. How do actuaries measure the unknown when the entire empty room of possible futures is literally being demolished and redrawn around them while they are trying to run the calculations? >> That is the ultimate existential challenge facing the future of risk modeling. How do you quantify an environment that has lost its historical tether? >> It is something for all of us, but especially those of you looking to pioneer the next generation of risk management to really think about as you study these foundational axioms. The math is perfect, but the reality it measures is moving under our feet. Thank you so much for joining us on this deep dive into the mathematics of uncertainty. >> Thank you for listening. Keep questioning the parameters of your reality and always respect the base rate.