# Expected Value

> Content from Expected Value.

## Pillar topics

- [Cognitive Biases](https://expectedvalue.co.uk/category/cognitive-biases/): Your brain is wired to misjudge probability. Anchoring, availability bias, the gamblers fallacy, sunk cost thinking — these mental shortcuts helped our ancestors survive but they consistently lead to poor decisions in modern life. Learn to recognise these biases in yourself and others, and discover practical techniques to counteract them.
- [Fundamentals](https://expectedvalue.co.uk/category/fundamentals/): Master the building blocks of probabilistic thinking. From Bayes theorem and expected value calculations to base rates and conditional probability, these guides lay the mathematical and conceptual foundation you need to make better decisions under uncertainty. Whether you are new to probability or brushing up, start here.
- [Investing](https://expectedvalue.co.uk/category/investing/): Markets are probability machines. Every investment is a bet on an uncertain future, and the best investors think in expected values, not certainties. Explore how probabilistic thinking applies to portfolio construction, risk management, index investing, and avoiding the behavioural traps that cost most investors dearly.
- [Strategy](https://expectedvalue.co.uk/category/strategy/): From career choices to business decisions, strategy is applied probability. Learn decision frameworks like expected value analysis, scenario planning, pre-mortems, and Kelly criterion sizing. These practical guides help you structure complex decisions and make consistently better choices when the stakes are high.
- [Tools & Resources](https://expectedvalue.co.uk/category/tools-resources/): Practical tools and resources for applying probabilistic thinking in your daily life. Calculators, spreadsheets, book recommendations, courses, and software that help you quantify uncertainty, track your predictions, and improve your calibration over time.

## Guides and articles

- [How Insurance Companies Use Probability to Make Money](https://expectedvalue.co.uk/blog/how-insurance-companies-use-probability/): 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.
- [When Is Travel Insurance Worth It? The EV Case](https://expectedvalue.co.uk/blog/travel-insurance-expected-value/): 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.
- [Expected Value Thinking: Better Decisions Under Uncertainty](https://expectedvalue.co.uk/blog/expected-value-explained/): Expected value thinking: the most important concept in decision-making under uncertainty. What it is, how to calculate it, when to apply it.
- [Nassim Taleb's Ideas: Black Swan, Antifragile & More](https://expectedvalue.co.uk/blog/nassim-taleb-ideas-explained/): 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.
- [Thinking in Bets: Decisions Like a Poker Pro](https://expectedvalue.co.uk/blog/thinking-in-bets-annie-duke/): Annie Duke's Thinking in Bets reframes every decision as a bet under uncertainty. The ideas of resulting, calibration, and decision groups.
- [Expected Value Calculator: How to Calculate EV Step-by-Step](https://expectedvalue.co.uk/blog/expected-value-calculator/): Learn how to calculate expected value with a clear formula, four worked examples (betting, investing, insurance, career) and the common mistakes to avoid.
- [Monte Carlo Thinking: How to Stress-Test Decisions](https://expectedvalue.co.uk/blog/monte-carlo-thinking/): Monte Carlo simulation lets you stress-test decisions across thousands of scenarios. A practical guide to using it for retirement, projects and investing.
- [Expected Value vs Expected Utility: When EV Isn't Enough](https://expectedvalue.co.uk/blog/expected-value-vs-expected-utility/): 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.
- [Hindsight Bias: Why Everything Looks Obvious After the Fact](https://expectedvalue.co.uk/blog/hindsight-bias/): Hindsight bias makes the past feel inevitable. Here's why it distorts post-mortems, juries and investing - and how to fight it.
- [Ergodicity Explained: Why Time Averages Matter Most](https://expectedvalue.co.uk/blog/ergodicity-explained/): Ergodicity separates sensible bets from catastrophic ones. The difference between ensemble averages and time averages - and why ignoring it can ruin you.
- [Conditional Probability Explained Simply, With Examples](https://expectedvalue.co.uk/blog/conditional-probability-explained/): Conditional probability is the chance one event happens given another already has. Worked examples in medical testing, weather, and the Monty Hall problem.
- [The Prosecutor's Fallacy: How Courts Get Statistics Wrong](https://expectedvalue.co.uk/blog/prosecutors-fallacy/): Confusing P(evidence | innocent) with P(innocent | evidence) sends innocent people to prison. The Sally Clark case and how to spot the trap.
- [The Dunning-Kruger Effect: Why We Overestimate Our Abilities](https://expectedvalue.co.uk/blog/dunning-kruger-effect/): 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.
- [The Law of Large Numbers: Why Casinos Always Win](https://expectedvalue.co.uk/blog/law-of-large-numbers/): 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.
- [The Availability Heuristic: Why Vivid Examples Mislead](https://expectedvalue.co.uk/blog/availability-heuristic/): Why we judge risk by what comes to mind first - fear of flying, market news, terrorism - and the practical techniques to debias your thinking.
- [Probability Calibration: Predict Like a Superforecaster](https://expectedvalue.co.uk/blog/probability-calibration-training/): Learn probability calibration like a superforecaster - Brier scores, drills, and free tools to sharpen your forecasts. Based on Philip Tetlock's research.
- [Sequence-of-Returns Risk: Why the Order Matters](https://expectedvalue.co.uk/blog/sequence-of-returns-risk/): Same average return, wildly different outcomes: why the ORDER of your returns decides how long a pension pot lasts. Sequence-of-returns risk explained.
- [Voluntary Excess: How to Choose the Level That Pays Off](https://expectedvalue.co.uk/blog/insurance-excess-expected-value/): Raising your voluntary excess cuts your premium but raises claim costs. Use expected value to find the excess level that actually pays off.
- [Are Extended Warranties Ever Worth It? The Expected Value](https://expectedvalue.co.uk/blog/extended-warranties-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.
- [What Investment Platform Fees Really Cost Over 20 Years](https://expectedvalue.co.uk/blog/investment-platform-fees-long-term-cost/): 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.
- [Premium Bonds: What's the Real Expected Value?](https://expectedvalue.co.uk/blog/premium-bonds-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.
- [UK FIRE Movement 2026: The Complete Roadmap](https://expectedvalue.co.uk/blog/uk-fire-movement-2026/): 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.
- [Stocks and Shares ISA Limit 2026: Rules and 2027 Changes](https://expectedvalue.co.uk/blog/stocks-and-shares-isa-limit-2026/): 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.
- [Where Can You Actually Find Positive Expected Value?](https://expectedvalue.co.uk/blog/where-to-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.
- [Availability Cascade in Markets](https://expectedvalue.co.uk/blog/availability-cascade-in-markets/): Availability cascade in markets: how compounding information cascades amplify trends, create bubbles, and drive systematic mispricing.
- [Mental Accounting Fallacy Explained](https://expectedvalue.co.uk/blog/mental-accounting-fallacy-explained/): Mental accounting fallacy: separating money into psychological 'buckets' distorts financial decisions; budgeting + investment implications.
- [Conjunction Fallacy in Business Decisions](https://expectedvalue.co.uk/blog/conjunction-fallacy-business-decisions/): Conjunction fallacy in business: how compound probability errors distort startup planning, investment scenarios, project timelines.
- [Decision Making Under Uncertainty Frameworks](https://expectedvalue.co.uk/blog/decision-making-uncertainty-frameworks/): Decision making under uncertainty frameworks: minimax regret, expected utility, satisficing, maximax compared. When each applies.
- [Expected Value in Poker Tournaments](https://expectedvalue.co.uk/blog/expected-value-poker-tournaments/): EV in poker tournaments: ICM considerations, bubble dynamics, when raw chip-EV diverges from actual tournament-EV.
- [Probability Weighted Utility for Decisions](https://expectedvalue.co.uk/blog/probability-weighted-utility-decisions/): Probability weighted utility for high-stakes decisions: combining probabilities and personal values for better choices.
- [Risk Aversion vs Loss Aversion Explained](https://expectedvalue.co.uk/blog/risk-aversion-vs-loss-aversion-explained/): Risk aversion vs loss aversion: the behavioural distinction; investment + portfolio implications; how each affects decisions.
- [Bayesian Updating in Everyday Life UK 2026](https://expectedvalue.co.uk/blog/bayesian-updating-everyday-life-uk-2026/): Bayesian updating UK 2026: practical examples of updating beliefs with new evidence in investment, health, and career decisions.
- [The Monty Hall Problem: Why You Should Always Switch](https://expectedvalue.co.uk/blog/monty-hall-problem-explained/): 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.
- [Survivorship Bias: The Hidden Data That Changes Everything](https://expectedvalue.co.uk/blog/survivorship-bias/): Survivorship bias hides the failures behind every success story - from WWII bombers to mutual funds. How to spot the missing data and decide better.
- [Overconfidence Bias: Why Active Traders Underperform](https://expectedvalue.co.uk/blog/overconfidence-bias/): Overconfidence bias is investing's most expensive cognitive error. How overestimation, the planning fallacy and overprecision hide it, and what fixes it.
- [12 Cognitive Biases That Wreck Probability Estimates](https://expectedvalue.co.uk/blog/cognitive-biases-probability/): The 12 cognitive biases that most consistently distort probability judgements - what each one does and how to defend against it.
- [Confirmation Bias in Investing: Fight Your Own Brain](https://expectedvalue.co.uk/blog/confirmation-bias-investing/): Confirmation bias quietly destroys investment returns. Here's how it works in markets - and the three techniques that actually neutralise it.
- [Regression to the Mean: Why Extremes Don't Last](https://expectedvalue.co.uk/blog/regression-to-the-mean/): Why extreme performance - sporting peaks, market gains, viral hits - almost always reverts to average. Galton's discovery, examples, and how to spot it.
- [Status Quo Bias: Why We Stick with Defaults](https://expectedvalue.co.uk/blog/status-quo-bias/): Status quo bias explained: the cognitive tendency to stick with defaults. UK financial impact: pension funds, insurance renewals, current accounts.
- [Black Swan Events: How to Prepare for the Unpredictable](https://expectedvalue.co.uk/blog/black-swan-events/): Black swan events explained: Taleb's definition, examples (2008, COVID, AI surge), and the barbell strategy + antifragility approach.
- [Probabilistic Thinking in Daily Life: 7 Practical Uses](https://expectedvalue.co.uk/blog/probabilistic-thinking-everyday-life/): How to apply probabilistic thinking to medical decisions, career bets, insurance, investing, dating, sports betting and travel. 7 worked examples.
- [Recency Bias in Investing and Sports Predictions](https://expectedvalue.co.uk/blog/recency-bias/): 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.
- [Decision Journals: Track Your Thinking to Improve It](https://expectedvalue.co.uk/blog/decision-journals/): 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.
- [Framing Effect: Why How You Hear It Changes Your Choice](https://expectedvalue.co.uk/blog/framing-effect-bias/): The framing effect: identical numbers presented differently produce opposite decisions. Asian Disease Problem, gain vs loss framing, defences.
- [Endowment Effect Explained: Why You Overvalue What You Own](https://expectedvalue.co.uk/blog/endowment-effect/): 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.
- [Negative Expected Value: Why Lottery Tickets Always Lose](https://expectedvalue.co.uk/blog/negative-expected-value/): Why every lottery ticket, roulette spin and slot pull is mathematically a loss in expectation. The maths behind why the house always wins.
- [Probability vs Odds: The Difference and Why It Matters](https://expectedvalue.co.uk/blog/probability-vs-odds/): Probability and odds describe the same uncertainty differently - and confusing them costs people money. How each works, conversions, and bookmaker tricks.
- [Fractional Kelly: Why Half Kelly Is the Smart Default](https://expectedvalue.co.uk/blog/fractional-kelly/): 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.
- [12 Best Books on Probabilistic Thinking and Decision-Making](https://expectedvalue.co.uk/blog/best-books-probabilistic-thinking-decision-making/): A curated reading list on probability, decision-making under uncertainty, and rational thinking - from Kahneman to practical guides for forecasters.
- [15 Best Decision Making Books to Change How You Think](https://expectedvalue.co.uk/blog/best-decision-making-books/): 15 best decision-making books - choice architecture, behavioural economics, frameworks, and habits. Mini-reviews, who each is for, and reading order.
- [Expected Value Formula: Derivation and 5 Worked Examples](https://expectedvalue.co.uk/blog/expected-value-formula/): The expected value formula is E[X] = Σ(p × x). Full derivation and five worked examples - coin flip, lottery, insurance, poker, and stock investment.
- [Kelly Criterion Calculator: Betting, Investing, Poker](https://expectedvalue.co.uk/blog/kelly-criterion-calculator/): Step-by-step Kelly Criterion calculator with worked examples for sports betting, investing and poker. Convert odds to stake percentages instantly.
- [Bayesian vs Frequentist Statistics: What's the Difference?](https://expectedvalue.co.uk/blog/bayesian-vs-frequentist/): Bayesian vs frequentist statistics - the philosophical split that decides whether you get p-values or posteriors, and when each one actually wins.
- [Bayes Theorem Explained: Formula and 5 Worked Examples](https://expectedvalue.co.uk/blog/bayes-theorem-explained/): Bayes theorem explained from the formula up: derivation, intuitive examples (medical tests, spam filters), and the base-rate trap that fools experts.
- [The Pre-Mortem - Imagine Your Decision Already Failed](https://expectedvalue.co.uk/blog/pre-mortem-decision-making/): The pre-mortem flips the post-mortem on its head: imagine the decision has already failed, then work backwards to find what optimism hides.
- [Thinking, Fast and Slow: The Bias Book Worth the Hype](https://expectedvalue.co.uk/blog/thinking-fast-and-slow-kahneman/): 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.
- [Decision Trees: A Visual Framework for Complex Choices](https://expectedvalue.co.uk/blog/decision-trees-visual-framework/): 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.
- [Bayesian Thinking for Everyday Decisions](https://expectedvalue.co.uk/blog/bayesian-thinking-everyday-decisions/): Bayesian thinking - the art of changing your mind rationally. How to update beliefs with evidence, with examples from interviews, medicine, and investing.
- [The Gambler's Fallacy: Why You're Wrong About 'Due' Outcomes](https://expectedvalue.co.uk/blog/gamblers-fallacy/): 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.
- [Second-Order Thinking: How to See Around Corners](https://expectedvalue.co.uk/blog/second-order-thinking/): Most decisions are made on first-order effects, but second-order consequences are where the surprises live. A framework for seeing past the obvious.
- [Base Rate Neglect: Why Your Intuitions Are Wrong](https://expectedvalue.co.uk/blog/base-rate-neglect/): 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.
- [The Kelly Criterion: How to Size Your Bets Optimally](https://expectedvalue.co.uk/blog/kelly-criterion-optimal-bet-sizing/): 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.
- [The False Positive Paradox: Why Positive Tests Mislead](https://expectedvalue.co.uk/blog/false-positive-paradox/): 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.
- [Sunk Cost Fallacy: When to Quit and When to Persist](https://expectedvalue.co.uk/blog/sunk-cost-fallacy/): 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.
- [A Probabilistic Framework for Career Decisions](https://expectedvalue.co.uk/blog/probabilistic-framework-career-decisions/): Use expected value, scenario planning and Kelly-style sizing to evaluate job offers, career pivots and salary negotiations honestly.
- [Expected Value in Poker: Pot Odds, EV & Decisions](https://expectedvalue.co.uk/blog/expected-value-poker/): Expected value in poker - pot odds, implied odds, EV of bluffs and folds. Worked hand examples showing how the maths translates into better decisions.
- [Position Sizing: How Much to Bet When You Have an Edge](https://expectedvalue.co.uk/blog/position-sizing-kelly-criterion/): 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.
- [Loss Aversion: Why Losses Hurt Twice as Much](https://expectedvalue.co.uk/blog/loss-aversion/): 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.
- [Thinking in Probabilities: Why Your Brain Is Bad at Risk](https://expectedvalue.co.uk/blog/thinking-in-probabilities/): Your brain systematically misjudges probability. Learn the cognitive biases that distort risk perception and how to calibrate better.
- [Anchoring Bias: How First Numbers Hijack Judgement](https://expectedvalue.co.uk/blog/anchoring-bias/): Anchoring bias: why the first number you hear silently warps every estimate that follows - with research, examples, and debiasing strategies that work.
- [Risk vs Uncertainty: The Distinction That Matters](https://expectedvalue.co.uk/blog/risk-vs-uncertainty/): Risk is measurable; uncertainty is not. Confusing the two produces overconfident forecasts and brittle portfolios. Here's how to tell them apart.
- [How Prediction Markets Work: Probability Meets Money](https://expectedvalue.co.uk/blog/how-prediction-markets-work/): Prediction markets turn questions about the future into tradeable contracts whose prices behave like probabilities. How they work, and how to use them.
- [Correlation vs Causation: A Probabilistic Thinking Guide](https://expectedvalue.co.uk/blog/correlation-vs-causation/): 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.

## Reviews

- [The Signal and the Noise Review (Nate Silver, 2012)](https://expectedvalue.co.uk/review/signal-and-noise-silver-review/): Research-led review of Nate Silver's The Signal and the Noise (2012). Bayesian reasoning, domain-by-domain forecastability, the 2016 misfire.
- [Fooled by Randomness Review (Nassim Taleb, 2001)](https://expectedvalue.co.uk/review/fooled-by-randomness-taleb-review/): Research-led review of Taleb's Fooled by Randomness (2001) - the foundational Incerto book on survivorship bias and the role of luck in success.
- [Superforecasting Review (Tetlock & Gardner, 2015)](https://expectedvalue.co.uk/review/superforecasting-tetlock-gardner-review/): Research-led review of Tetlock & Gardner's Superforecasting (2015). Good Judgment Project + the 10-commandment framework + an honest read of the limits.

## Comparisons

- [Vanguard vs InvestEngine: UK 2026 Comparison](https://expectedvalue.co.uk/compare/vanguard-vs-investengine-uk-2026/): InvestEngine charges no platform fee, so it is cheaper for small UK portfolios; Vanguard suits LifeStrategy fans and large pots under its fee cap.
- [Best SIPP UK 2026: 10 Providers Compared](https://expectedvalue.co.uk/compare/best-sipp-uk-2026/): Compare the 10 best UK SIPP providers for 2026 on fees, dealing charges and drawdown - AJ Bell, ii, Vanguard, InvestEngine, HL and more.
- [Kalshi vs Polymarket vs Manifold (UK & US 2026)](https://expectedvalue.co.uk/compare/kalshi-vs-polymarket-vs-manifold/): Kalshi is regulated real money but excludes the UK; Polymarket is crypto-settled and open to UK users; Manifold is play money. Which one fits you.
- [Trading 212 vs eToro: UK 2026 Comparison](https://expectedvalue.co.uk/compare/trading-212-vs-etoro-uk-2026/): Trading 212 is the cheaper, simpler ISA for most UK investors, with a low FX fee and no custody fee. Pick eToro only if you want copy trading. 2026.
- [Thinking, Fast and Slow vs Thinking in Bets](https://expectedvalue.co.uk/compare/thinking-fast-and-slow-vs-thinking-in-bets/): Kahneman explains how your brain decides; Annie Duke gives you tools to decide better. Which decision-making book to read first, and why to read both.

## Products

- [Superforecasting: The Art and Science of Prediction](https://expectedvalue.co.uk/product/superforecasting-tetlock-gardner/): Philip Tetlock + Dan Gardner's 2015 book on Good Judgment Project research into what makes forecasters more accurate. Foundational text.
- [The Signal and the Noise: Why Predictions Fail](https://expectedvalue.co.uk/product/signal-and-noise-nate-silver/): Nate Silver's 2012 book on probabilistic forecasting across weather, sports, poker, economics, politics. Builds the case for Bayesian reasoning.
- [Fooled by Randomness: The Hidden Role of Chance](https://expectedvalue.co.uk/product/fooled-by-randomness-taleb/): Nassim Taleb's 2001 book on luck in success and how survivorship bias distorts judgment of skill in finance + business. First Incerto text.

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