# Probability and uncertainty This opening slide frames probability as a practical tool for judgment when the full truth is hidden. It uses three familiar cases — investing, health decisions, and news interpretation — to show that uncertainty is not a special topic for casinos or textbooks, but a normal feature of everyday choices. The key idea is that good reasoning starts by resisting false certainty. Instead of collapsing each situation into one story, we ask what outcomes are possible, how likely they are, and what the consequences would be. The rest of the lesson turns that habit into concrete tools: <ref slide="2">Expected value</ref> for weighing payoffs, <ref slide="3">Bayes and base rates</ref> for updating beliefs, <ref slide="4">Compounding and growth</ref> for repeated change over time, <ref slide="5">Optimization and trade-offs</ref> for choosing under constraints, and <ref slide="6">Estimation and Fermi thinking</ref> for making grounded rough guesses. ## Expected value When a choice has multiple possible outcomes, **expected value** gives you a compact way to compare them. It is the average payoff you would expect if a similar situation happened many times. That does not mean it predicts the exact result of one trial; it helps you judge whether a bet, project, or offer is good on average. This is one of the most useful decision tools for a technical mind because it compresses uncertainty into something comparable. A side project with a small chance of a huge upside may beat a safe but low-upside option. An insurance product that feels comforting may still be a bad deal if the premium is consistently larger than the expected loss reduction. Expected value helps you see past emotional salience and into long-run structure. <viz id="1"></viz> **Pick an example** and **move the probability and payoff sliders**. **Watch the expected value update live** and **compare it with the alternative** as the ranking changes. Notice how your intuition can get pulled toward vivid outcomes: a big jackpot, a scary downside, a prestigious title. Expected value asks you to weight each outcome by how often it really happens, not by how emotionally loud it feels. Of course, real decisions are not only about averages. Risk tolerance, downside constraints, and time horizon still matter. But expected value is the baseline sanity check: before debating subtle preferences, first ask whether the option is even good on average. Once you have that habit, the next question is how to update those probabilities when new evidence arrives, which leads directly to <ref slide="3">Bayes and base rates</ref>. ## Bayes and base rates A common reasoning failure is to focus on a signal and ignore how common the underlying thing was to begin with. **Base rates** are those background frequencies. **Bayesian thinking** means updating your belief by combining the prior odds with the strength of the new evidence. This matters whenever you interpret alerts, tests, forecasts, dashboards, or suspicious-looking patterns. If something is rare, then even a pretty accurate signal can produce many false alarms. For a developer, this is similar to debugging with noisy monitoring: an alert is not just about model accuracy; it also depends on how often the underlying failure actually occurs. <viz id="2"></viz> **Adjust how common the condition is** and **change the signal accuracy**. **Focus on the positive results** and **observe how the share of real positives changes** even when the test seems accurate. The aha insight is that evidence does not speak in isolation. A positive signal is more convincing when the thing being tested is already somewhat plausible, and less convincing when it is extremely rare. That is why raw accuracy numbers can mislead if you do not also ask, “Accurate relative to what base rate?” This complements <ref slide="2">Expected value</ref>: expected value helps you decide once you have probabilities, and Bayes helps you improve those probabilities when you get new information. Another major judgment upgrade comes from understanding how small rates accumulate over time, which is the focus of <ref slide="4">Compounding and growth</ref>. ## Compounding and growth Humans are not naturally good at feeling the force of repeated percentage change. **Compounding** happens when growth in one period increases the base for growth in the next period. That creates curves that start quietly and then become dramatic. By contrast, **linear growth** adds the same amount each step. This idea shows up everywhere: savings, debt, software adoption, learning, and even repeated small improvements in habits or systems. A tiny rate difference can look trivial in the short run and decisive in the long run. That is why rates often matter more than single snapshots. <viz id="3"></viz> **Move the rate slider** and **extend the time horizon**. **Compare the linear and compound curves** and **notice how a small rate change creates a much larger final gap** over longer periods. The important judgment lesson is to pay attention to mechanisms that repeat. One-off gains are easy to notice, but recurring multipliers quietly dominate outcomes over time. That applies positively in investing and skill building, and negatively in debt, churn, or technical decay. Once you start seeing rates instead of isolated events, many real-life choices become clearer. But not every decision reduces to growth alone. Often you need to compare several competing dimensions at once, which is where <ref slide="5">Optimization and trade-offs</ref> becomes useful. ## Optimization and trade-offs Many choices feel hard not because they are mysterious, but because they involve multiple goals that conflict. **Optimization** means choosing the best option according to a defined objective. **Trade-offs** appear when improving one dimension makes another worse. This matters for decisions like jobs, apartments, tools, or product strategy. Higher salary may come with more stress. Lower rent may cost more commute time. A role with less immediate pay may offer stronger learning. The core lesson is that there is rarely a universally best option; there is a best option relative to what you value and how much you value it. <viz id="4"></viz> **Adjust the weights on each factor** and **watch the rankings change**. **Try making one factor dominate** and then **balance the weights more evenly** to see how the preferred option shifts. This is conceptually close to writing a scoring function in software: the output depends on the objective you encoded. If your priorities are fuzzy, your decisions will feel fuzzy too. Making weights explicit does not remove judgment, but it turns hidden assumptions into something inspectable. This also helps you combine earlier ideas. You may estimate probabilities with <ref slide="3">Bayes and base rates</ref>, compare uncertain payoffs with <ref slide="2">Expected value</ref>, and then optimize across several dimensions at once. To do all that well in messy situations, one more habit is invaluable: being able to estimate quickly even without exact data, which is the focus of <ref slide="6">Estimation and Fermi thinking</ref>. # Estimation and Fermi thinking This closing slide frames Fermi thinking as a practical reality-check tool for everyday decisions. It uses three familiar cases—travel time, server cost, and plausibility checks—to show how rough decomposition can outperform an unsupported gut feeling. The slide connects estimation back to earlier ideas in the lesson: uncertainty from <ref slide="1">Probability and uncertainty</ref>, scale effects from <ref slide="4">Compounding and growth</ref>, and focusing on the biggest drivers from <ref slide="5">Optimization and trade-offs</ref>. The key takeaway is that an order-of-magnitude answer is often enough to decide whether a claim looks reasonable, suspicious, or worth deeper analysis.