https://expectedvalue.co.uk https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/0c9b5b4f-6ff6-44a9-83c6-b4dbdf690a5b.jpg Hero illustration of overlapping probability distributions and decision branches, framing the site's focus on probabilistic thinking and rational decisions. https://expectedvalue.co.uk/about/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/c39e439d-8a38-40ff-bbc8-c58ce9f54495.jpg Editorial illustration of a thinker weighing options on a balance scale, suggesting deliberate, probability-aware reasoning. https://expectedvalue.co.uk/blog/anchoring-bias/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/297b5be7-871b-4bd3-8424-aca587df6084.jpg Initial number on a price tag visibly anchoring a downstream estimate, illustrating how first impressions hijack judgement. https://expectedvalue.co.uk/blog/base-rate-neglect/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/27fca488-f7fd-4cae-84c0-3ae9d723542a.jpg Illustration contrasting a small biased sample against the underlying base-rate population, with arrows highlighting the gap. https://expectedvalue.co.uk/blog/bayesian-thinking-everyday-decisions/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/43a4f84d-4db6-4056-8027-4380ddbd7e29.jpg Visual representation of prior and posterior probability distributions updating in light of new evidence. https://expectedvalue.co.uk/blog/best-books-probabilistic-thinking-decision-making/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/cfe420fa-200d-4386-bcde-30dc057a541c.jpg Stack of books on probabilistic thinking and decision theory arranged on a desk with a notebook and pen. https://expectedvalue.co.uk/blog/best-decision-making-books/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/open-decision-making-books-on-a-wooden-desk-under-warm-light-e1af4dc6.jpg Open decision-making books on a wooden desk under warm light https://expectedvalue.co.uk/blog/correlation-vs-causation/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/e1b027be-1bd1-430a-a021-dadf90f0e98c.jpg Scatter plot showing two correlated variables, with a question mark over the arrow that would imply causation. https://expectedvalue.co.uk/blog/dunning-kruger-effect/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/cd9eb39c-72f5-4bd8-91fe-ed1ecaebe4f2.jpg Curve showing confidence rising sharply at low competence and dipping in the middle, depicting the Dunning-Kruger pattern. https://expectedvalue.co.uk/blog/ergodicity-explained/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/a-long-term-financial-chart-showing-the-divergence-between-3f63195e.jpg A long-term financial chart showing the divergence between ensemble and time-averaged paths https://expectedvalue.co.uk/blog/expected-value-explained/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/304ec65d-0d74-49b9-89ad-9209d8e10ebc.jpg Decision tree branching into multiple outcomes with probabilities and payoffs annotated, illustrating expected-value calculation. https://expectedvalue.co.uk/blog/expected-value-poker/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/poker-chips-and-cards-1c533db4.jpg Poker chips and playing cards on a green felt table — calculating expected value at the felt https://expectedvalue.co.uk/blog/false-positive-paradox/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/2affa03c-e629-40bf-a196-b81c878c2d76.jpg Decision tree showing how a positive test result with low base rates produces a counter-intuitive number of false positives. https://expectedvalue.co.uk/blog/gamblers-fallacy/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/0bb5b5c6-655c-4484-93ba-b4b902a82661.jpg Sequence of coin flips with an arrow incorrectly suggesting a 'due' outcome, illustrating the gambler's fallacy in pattern perception. https://expectedvalue.co.uk/blog/hindsight-bias/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/910d7a65-288b-42f2-bd41-74b74c5fc7a9.jpg Backward-looking timeline where past events appear inevitable in retrospect, illustrating hindsight bias. https://expectedvalue.co.uk/blog/how-insurance-companies-use-probability/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/insurance-and-probability-actuarial-calculations-and-risk-7fbd5368.jpg Insurance and probability — actuarial calculations and risk assessment https://expectedvalue.co.uk/blog/how-prediction-markets-work/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/ffed4b94-f971-4c25-aad2-348438e3a28f.jpg Stylised order book translating contract prices into probability estimates for a real-world event. https://expectedvalue.co.uk/blog/kelly-criterion-optimal-bet-sizing/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/80e7cd52-4713-452c-aac8-cdead955e811.jpg Stylised growth curve showing optimal bet sizing under the Kelly criterion versus over-betting and under-betting trajectories. https://expectedvalue.co.uk/blog/loss-aversion/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/bdff1807-6bb2-4925-9b7d-67c48ea4deab.jpg Asymmetric value function curve showing pain of losses outweighing pleasure of equivalent gains, the central finding of prospect theory. https://expectedvalue.co.uk/blog/position-sizing-kelly-criterion/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/33424494-6901-4d89-921a-bc2b36b254f3.jpg Bar chart showing how bankroll allocation scales with edge, illustrating the position-sizing principle behind the Kelly criterion. https://expectedvalue.co.uk/blog/pre-mortem-decision-making/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/team-gathered-around-a-whiteboard-sketching-project-plans-8c977ade.jpg Team gathered around a whiteboard sketching project plans and risk scenarios https://expectedvalue.co.uk/blog/probabilistic-framework-career-decisions/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/3616399f-b802-4b7e-b635-96806a8632de.jpg Decision matrix weighing career options against payoffs and probabilities of success, modelling a probabilistic approach to career choice. https://expectedvalue.co.uk/blog/probability-vs-odds/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/83f20e8b-6a3b-4e8e-88d2-da0c0407e38a.jpg Side-by-side equations converting between probability and odds for the same event, showing where the formats agree and diverge. https://expectedvalue.co.uk/blog/risk-vs-uncertainty/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/5d9efc1d-c54c-422c-80eb-9a854ead624f.jpg Two distributions side by side: one with quantified risk and known probabilities, one with unbounded uncertainty. https://expectedvalue.co.uk/blog/second-order-thinking/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/0e9903c5-1125-42cc-8535-db9fe31b35f7.jpg Chain of consequences extending beyond a single decision, illustrating how second-order effects can outweigh the immediate one. https://expectedvalue.co.uk/blog/sunk-cost-fallacy/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/a729f9f4-189e-40b1-995b-e2b9ac0dd1ad.jpg Person at a fork in the road weighing past investment against future expected value, symbolising the sunk-cost decision. https://expectedvalue.co.uk/blog/thinking-in-probabilities/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/ac1b8277-33c8-431b-ab9f-d7761a532392.jpg Abstract illustration of overlapping probability distributions, suggesting how uncertain outcomes are best understood as ranges, not points. https://expectedvalue.co.uk/tools/ https://expectedvalue.co.uk/assets/5df1ee4f-c906-43dc-a9d0-c15a782662ce/acec120a-2c70-4af4-9d52-80769688d340.jpg Collection of decision-making tools — calculator, notebook, expected-value worksheet — laid out on a workspace.