Skip to main content
Expected Value
Start Here
Topics
FundamentalsCognitive BiasesStrategyInvesting
ToolsComparisons
Start Here Topics FundamentalsCognitive BiasesStrategyInvesting Tools Comparisons
  1. Home
  2. Search

Search

Search the archive.

79 entries · posts, reviews, products

Strategy

Black Swan Events: How to Prepare for the Unpredictable

Black swan events explained: Taleb's definition, examples (2008, COVID, AI surge), and the barbell strategy + antifragility approach.

2026-06-02T04:34:01.682Z
Cognitive Biases

Base Rate Neglect: Why Your Intuitions Are Wrong

Base rate neglect is one of the most costly cognitive biases. Why most positive medical tests are wrong, and how to train yourself to think in base rates.

Cognitive Biases

Loss Aversion: Why Losses Hurt Twice as Much

Loss aversion: why losses hurt twice as much as equivalent gains feel good. The Kahneman/Tversky research, where it warps decisions, and how to counter it.

Cognitive Biases

Status Quo Bias: Why We Stick with Defaults

Status quo bias explained: the cognitive tendency to stick with defaults. UK financial impact: pension funds, insurance renewals, current accounts.

2026-06-02T04:36:19.114Z
Fundamentals

How Prediction Markets Work: Probability Meets Money

Prediction markets turn questions about the future into tradeable contracts whose prices behave like probabilities. How they work, and how to use them.

Cognitive Biases

Thinking, Fast and Slow: The Bias Book Worth the Hype

Kahneman's Thinking, Fast and Slow is the canonical text on cognitive bias. What it gets right, what hasn't aged well, and whether to buy it.

2026-05-13T21:26:02.672Z
Fundamentals

How Insurance Companies Use Probability to Make Money

Insurance is a negative-EV bet - for you. Here's how actuarial science, risk pooling, and the law of large numbers turn that into reliable profit.

Investing

When Is Travel Insurance Worth It? The EV Case

Travel insurance is usually negative-EV, yet often the right call. The expected-value rule for when a risk is worth insuring - and when to self-insure.

2026-07-25T11:09:36.925Z
Investing

Sequence-of-Returns Risk: Why the Order Matters

Same average return, wildly different outcomes: why the ORDER of your returns decides how long a pension pot lasts. Sequence-of-returns risk explained.

2026-07-12T16:45:35.392Z
Fundamentals

Expected Value Thinking: Better Decisions Under Uncertainty

Expected value thinking: the most important concept in decision-making under uncertainty. What it is, how to calculate it, when to apply it.

Tools & Resources

15 Best Decision Making Books to Change How You Think

15 best decision-making books - choice architecture, behavioural economics, frameworks, and habits. Mini-reviews, who each is for, and reading order.

Fundamentals

Nassim Taleb's Ideas: Black Swan, Antifragile & More

A guide to Nassim Taleb's key ideas - Black Swans, Antifragile, Skin in the Game, fat tails, barbell strategy and the Lindy effect - and where to start.

Strategy

Thinking in Bets: Decisions Like a Poker Pro

Annie Duke's Thinking in Bets reframes every decision as a bet under uncertainty. The ideas of resulting, calibration, and decision groups.

Fundamentals

12 Best Books on Probabilistic Thinking and Decision-Making

A curated reading list on probability, decision-making under uncertainty, and rational thinking - from Kahneman to practical guides for forecasters.

Fundamentals

Probabilistic Thinking in Daily Life: 7 Practical Uses

How to apply probabilistic thinking to medical decisions, career bets, insurance, investing, dating, sports betting and travel. 7 worked examples.

2026-06-02T01:42:14.329Z
Investing

Position Sizing: How Much to Bet When You Have an Edge

Knowing you have an edge is half the battle - sizing positions correctly is the other half. How to size bets so you don't blow up or waste your advantage.

Strategy

Expected Value in Poker: Pot Odds, EV & Decisions

Expected value in poker - pot odds, implied odds, EV of bluffs and folds. Worked hand examples showing how the maths translates into better decisions.

Tools & Resources

Expected Value Calculator: How to Calculate EV Step-by-Step

Learn how to calculate expected value with a clear formula, four worked examples (betting, investing, insurance, career) and the common mistakes to avoid.

Fundamentals

Monte Carlo Thinking: How to Stress-Test Decisions

Monte Carlo simulation lets you stress-test decisions across thousands of scenarios. A practical guide to using it for retirement, projects and investing.

Fundamentals

Expected Value Formula: Derivation and 5 Worked Examples

The expected value formula is E[X] = Σ(p × x). Full derivation and five worked examples - coin flip, lottery, insurance, poker, and stock investment.

2026-05-23T16:56:29.025Z
Strategy

Fractional Kelly: Why Half Kelly Is the Smart Default

Full Kelly is mathematically optimal but emotionally brutal. Why half Kelly is the practical default, when to scale up or down, and how the pros use it.

2026-05-28T10:27:35.398Z
Cognitive Biases

The Gambler's Fallacy: Why You're Wrong About 'Due' Outcomes

The Gambler's Fallacy: why we wrongly believe random outcomes are 'due'. The cognitive trap, where it costs you, and how to think clearly under randomness.

Fundamentals

Expected Value vs Expected Utility: When EV Isn't Enough

Pure expected value can lead to ruinous decisions. Here's why expected utility, risk aversion, and the St Petersburg paradox matter for real-life choices.

Cognitive Biases

12 Cognitive Biases That Wreck Probability Estimates

The 12 cognitive biases that most consistently distort probability judgements - what each one does and how to defend against it.

2026-06-10T17:45:36.382Z
Strategy

Decision Trees: A Visual Framework for Complex Choices

Decision trees turn tangled choices into a diagram you can actually solve. Build one in five steps, work through three examples, and avoid common traps.

2026-05-13T21:22:57.454Z
Cognitive Biases

Hindsight Bias: Why Everything Looks Obvious After the Fact

Hindsight bias makes the past feel inevitable. Here's why it distorts post-mortems, juries and investing - and how to fight it.

Fundamentals

Ergodicity Explained: Why Time Averages Matter Most

Ergodicity separates sensible bets from catastrophic ones. The difference between ensemble averages and time averages - and why ignoring it can ruin you.

Fundamentals

Conditional Probability Explained Simply, With Examples

Conditional probability is the chance one event happens given another already has. Worked examples in medical testing, weather, and the Monty Hall problem.

Cognitive Biases

The Prosecutor's Fallacy: How Courts Get Statistics Wrong

Confusing P(evidence | innocent) with P(innocent | evidence) sends innocent people to prison. The Sally Clark case and how to spot the trap.

Cognitive Biases

The Dunning-Kruger Effect: Why We Overestimate Our Abilities

The Dunning-Kruger effect: the gap between how good people think they are and how good they actually are. The real research is more useful than the chart.

Cognitive Biases

Bayesian Thinking for Everyday Decisions

Bayesian thinking - the art of changing your mind rationally. How to update beliefs with evidence, with examples from interviews, medicine, and investing.

Fundamentals

Risk vs Uncertainty: The Distinction That Matters

Risk is measurable; uncertainty is not. Confusing the two produces overconfident forecasts and brittle portfolios. Here's how to tell them apart.

Fundamentals

The Law of Large Numbers: Why Casinos Always Win

The Law of Large Numbers explains why a single bet is wildly volatile but casinos profit reliably - the maths behind insurance and long-run averages.

Cognitive Biases

Anchoring Bias: How First Numbers Hijack Judgement

Anchoring bias: why the first number you hear silently warps every estimate that follows - with research, examples, and debiasing strategies that work.

Fundamentals

Decision Making Under Uncertainty Frameworks

Decision making under uncertainty frameworks: minimax regret, expected utility, satisficing, maximax compared. When each applies.

2026-06-15T16:46:44.074Z
Cognitive Biases

Mental Accounting Fallacy Explained

Mental accounting fallacy: separating money into psychological 'buckets' distorts financial decisions; budgeting + investment implications.

2026-06-15T16:50:57.600Z
Cognitive Biases

The Availability Heuristic: Why Vivid Examples Mislead

Why we judge risk by what comes to mind first - fear of flying, market news, terrorism - and the practical techniques to debias your thinking.

Fundamentals

Negative Expected Value: Why Lottery Tickets Always Lose

Why every lottery ticket, roulette spin and slot pull is mathematically a loss in expectation. The maths behind why the house always wins.

2026-05-31T23:52:18.222Z
Strategy

A Probabilistic Framework for Career Decisions

Use expected value, scenario planning and Kelly-style sizing to evaluate job offers, career pivots and salary negotiations honestly.

Fundamentals

Probability vs Odds: The Difference and Why It Matters

Probability and odds describe the same uncertainty differently - and confusing them costs people money. How each works, conversions, and bookmaker tricks.

Tools & Resources

Kelly Criterion Calculator: Betting, Investing, Poker

Step-by-step Kelly Criterion calculator with worked examples for sports betting, investing and poker. Convert odds to stake percentages instantly.

2026-05-23T16:53:52.436Z
Fundamentals

The False Positive Paradox: Why Positive Tests Mislead

When a test for a rare condition comes back positive, it's often more likely wrong than right. The false positive paradox, explained with real numbers.

Investing

What Investment Platform Fees Really Cost Over 20 Years

A 0.35% platform fee sounds tiny. Compounded over 25 years it can cost £30,000+. The maths of fee drag, and where flat fees beat percentages.

2026-07-01T22:13:39.814Z
Fundamentals

Probability Weighted Utility for Decisions

Probability weighted utility for high-stakes decisions: combining probabilities and personal values for better choices.

2026-06-15T16:43:34.377Z
Fundamentals

Bayesian Updating in Everyday Life UK 2026

Bayesian updating UK 2026: practical examples of updating beliefs with new evidence in investment, health, and career decisions.

2026-06-15T16:40:32.027Z
Cognitive Biases

Conjunction Fallacy in Business Decisions

Conjunction fallacy in business: how compound probability errors distort startup planning, investment scenarios, project timelines.

2026-06-15T16:48:17.273Z
Cognitive Biases

Sunk Cost Fallacy: When to Quit and When to Persist

The sunk cost fallacy: why we keep investing in losing decisions. The bias, the psychology, and a clear test for when to quit and when to persist.

Cognitive Biases

Risk Aversion vs Loss Aversion Explained

Risk aversion vs loss aversion: the behavioural distinction; investment + portfolio implications; how each affects decisions.

2026-06-15T16:42:01.432Z
Tools & Resources

Second-Order Thinking: How to See Around Corners

Most decisions are made on first-order effects, but second-order consequences are where the surprises live. A framework for seeing past the obvious.

Strategy

Expected Value in Poker Tournaments

EV in poker tournaments: ICM considerations, bubble dynamics, when raw chip-EV diverges from actual tournament-EV.

2026-06-15T16:45:09.510Z
Cognitive Biases

Thinking in Probabilities: Why Your Brain Is Bad at Risk

Your brain systematically misjudges probability. Learn the cognitive biases that distort risk perception and how to calibrate better.

Fundamentals

Correlation vs Causation: A Probabilistic Thinking Guide

Correlation is not causation - the most-quoted line in statistics, and the most misunderstood. What it really means, and how to think clearly about cause.

Strategy

Probability Calibration: Predict Like a Superforecaster

Learn probability calibration like a superforecaster - Brier scores, drills, and free tools to sharpen your forecasts. Based on Philip Tetlock's research.

Strategy

UK FIRE Movement 2026: The Complete Roadmap

A UK FIRE roadmap for 2026: the 4% rule, your real number, the tax-wrapper order, and the ISA bridge to pension access age. Honest maths, no hype.

2026-06-27T22:30:24.964Z
Investing

Are Extended Warranties Ever Worth It? The Expected Value

Warranty maths retailers hope you never do: premium vs failure odds x repair cost. Why most UK cover is -EV, and the one case where it isn't.

2026-07-01T22:18:39.049Z
Investing

Voluntary Excess: How to Choose the Level That Pays Off

Raising your voluntary excess cuts your premium but raises claim costs. Use expected value to find the excess level that actually pays off.

2026-07-11T13:20:53.518Z
Fundamentals

The Kelly Criterion: How to Size Your Bets Optimally

The Kelly Criterion tells you exactly how much to stake when you have an edge. Formula derivation, fractional Kelly, history, and worked examples for 2026.

Investing

Premium Bonds: What's the Real Expected Value?

Premium Bonds advertise 3.8%, but that average hides a skewed reality. Here's the real expected (and typical) return for £1k, £10k and £50k holdings.

2026-06-27T22:35:26.181Z
Strategy

Where Can You Actually Find Positive Expected Value?

Almost every bet is negative-EV by design. Here are the few places an ordinary person can genuinely find positive expected value.

2026-06-26T17:18:26.498Z
Investing

Stocks and Shares ISA Limit 2026: Rules and 2027 Changes

The 2026/27 stocks and shares ISA limit is £20,000. Here's how the allowance works, what the April 2027 cash ISA cut means, and how to use it.

2026-06-27T22:24:29.564Z
Fundamentals

Bayes Theorem Explained: Formula and 5 Worked Examples

Bayes theorem explained from the formula up: derivation, intuitive examples (medical tests, spam filters), and the base-rate trap that fools experts.

2026-05-21T10:08:48.077Z
Cognitive Biases

Framing Effect: Why How You Hear It Changes Your Choice

The framing effect: identical numbers presented differently produce opposite decisions. Asian Disease Problem, gain vs loss framing, defences.

2026-06-01T20:11:59.835Z
Cognitive Biases

Availability Cascade in Markets

Availability cascade in markets: how compounding information cascades amplify trends, create bubbles, and drive systematic mispricing.

2026-06-15T16:52:36.355Z
Fundamentals

The Monty Hall Problem: Why You Should Always Switch

The Monty Hall problem looks 50/50 and isn't - switching doors wins two-thirds of the time. Here's why, with five proofs and the famous controversy.

Fundamentals

Bayesian vs Frequentist Statistics: What's the Difference?

Bayesian vs frequentist statistics - the philosophical split that decides whether you get p-values or posteriors, and when each one actually wins.

2026-05-21T10:12:17.300Z
Cognitive Biases

Recency Bias in Investing and Sports Predictions

Recency bias: why investors chase last quarter's winners and bettors back the form team. The mechanics, the cost, and the four tools that defuse it.

2026-06-01T23:49:32.642Z
Cognitive Biases

Survivorship Bias: The Hidden Data That Changes Everything

Survivorship bias hides the failures behind every success story - from WWII bombers to mutual funds. How to spot the missing data and decide better.

Cognitive Biases

Overconfidence Bias: Why Active Traders Underperform

Overconfidence bias is investing's most expensive cognitive error. How overestimation, the planning fallacy and overprecision hide it, and what fixes it.

Cognitive Biases

Confirmation Bias in Investing: Fight Your Own Brain

Confirmation bias quietly destroys investment returns. Here's how it works in markets - and the three techniques that actually neutralise it.

Fundamentals

Regression to the Mean: Why Extremes Don't Last

Why extreme performance - sporting peaks, market gains, viral hits - almost always reverts to average. Galton's discovery, examples, and how to spot it.

Strategy

The Pre-Mortem - Imagine Your Decision Already Failed

The pre-mortem flips the post-mortem on its head: imagine the decision has already failed, then work backwards to find what optimism hides.

Cognitive Biases

Endowment Effect Explained: Why You Overvalue What You Own

The endowment effect makes you value the things you own more highly than identical things you do not. Here is why it happens, with worked examples.

2026-06-01T11:14:09.937Z
Strategy

Decision Journals: Track Your Thinking to Improve It

A decision journal is the only honest record of how well you actually think. What to record, when to review, and the mistakes that ruin the practice.

Product

Superforecasting: The Art and Science of Prediction

Philip Tetlock + Dan Gardner's 2015 book on Good Judgment Project research into what makes forecasters more accurate. Foundational text.

Product

Fooled by Randomness: The Hidden Role of Chance

Nassim Taleb's 2001 book on luck in success and how survivorship bias distorts judgment of skill in finance + business. First Incerto text.

Product

The Signal and the Noise: Why Predictions Fail

Nate Silver's 2012 book on probabilistic forecasting across weather, sports, poker, economics, politics. Builds the case for Bayesian reasoning.

Review

The Signal and the Noise Review (Nate Silver, 2012)

<p>Nate Silver's The Signal and the Noise sits between Taleb's <em>Fooled by Randomness</em> (diagnostic) and Tetlock's <em>Superforecasting</em> (prescriptive). Where Taleb argues that most forecasting is bunk and Tetlock argues that calibration is learnable, Silver argues that forecastability varies dramatically across domains - weather is genuinely predictable to a useful horizon, terrorism is not, and most everyday economic and political forecasting sits somewhere in between.</p><p>The book's biggest contribution is making Bayesian reasoning accessible to readers without statistical training. Silver works through Bayes' theorem with concrete examples (the cancer-screening case, the lab-test case, the baseball PECOTA forecasting case) in a way that demystifies the maths without dumbing it down.</p><p>Two caveats. First, the 2012 publication date predates Silver's 2016 election misfire, which raised hard questions about his confidence in the political-prediction sections. The book's claims are mostly defensible - 2016 was a rare-event outcome at the edge of the predicted distribution - but reading it without that context misses an important corrective. Second, the prose is more data-journalism-flat than Kahneman or Taleb; readers who prefer punchy aphorism may bounce.</p><p>Recommended as the practical applications complement to Tetlock + Taleb.</p>

2026-06-02T01:49:45.558Z
Review

Fooled by Randomness Review (Nassim Taleb, 2001)

<p>Fooled by Randomness is the book that established Nassim Taleb's intellectual brand. Published in 2001, it makes the case that success in finance, business, and other competitive domains is far more shaped by luck than the people involved (or the journalists who write about them) acknowledge. Survivorship bias is the central organising idea: the fund manager who beat the market for ten years looks impressive until you remember the 999 similar managers who didn't survive to be interviewed.</p><p>The book is readable, often funny, and rich in market-anecdote case studies. Twenty-five years on, its central ideas have been so widely absorbed that newer readers may find the argument obvious - which is exactly the kind of compliment Taleb's books pay themselves into. Where it falls short is in quantitative method: the conceptual frame is sharp but the practical 'what should I do differently' content is thinner than in <em>The Black Swan</em> (2007), the next book in the Incerto series.</p><p>For most readers, Fooled by Randomness is the right Taleb entry point. It's shorter, more accessible, and frames the survivorship-and-luck core that the rest of the Incerto develops further. Recommended.</p>

2026-06-02T01:47:05.121Z
Review

Superforecasting Review (Tetlock & Gardner, 2015)

<p>Superforecasting is the most rigorous book in the probabilistic-thinking canon. Unlike most decision-making books that synthesise existing research into a narrative, Superforecasting reports on a specific 4-year research programme - the Good Judgment Project, funded by IARPA - that competitively measured what makes some forecasters dramatically more accurate than others. The findings are practical, actionable, and grounded in data rather than anecdote.</p><p>The book's central claim is that calibration in short-horizon forecasting (months to 18 months out, on political and economic questions) is a learnable skill rather than a fixed trait. The 'superforecasters' - the top 2% of the Good Judgment Project's volunteer forecaster pool - shared a specific set of habits that the rest of us can practise. That makes Superforecasting useful in a way Kahneman's <em>Thinking, Fast and Slow</em> (more diagnostic) and Taleb's <em>Fooled by Randomness</em> (more philosophical) aren't.</p><p>Where it falls short: the narrative-heavy first third can feel slow before the practical material lands, and the findings explicitly cap at ~12-18 month horizons - this isn't a guide to long-range forecasting. For most readers, neither caveat materially detracts. Recommended.</p>

2026-06-02T01:44:39.556Z

No matches. Try a different word.

Expected Value

Better decisions through probabilistic thinking

Topics

Start HereFundamentalsCognitive BiasesInvestingStrategy

Site

AboutTools & ResourcesContactPrivacy PolicyCorrections
© 2026 Expected Value expectedvalue.co.uk
Cookies

Your privacy choices

We use cookies for essential site functionality and, with your consent, for analytics. You can change your choice at any time from the footer. Privacy policy