Thinking in Distributions, Not Points

## The Single-Point Prediction Problem Most bettors think in single-point predictions: "Arsenal will score 2 goals." "The match will end 1-0." These point predictions ignore the crucial fact that sports outcomes are probabilistic — there is a distribution of possible outcomes, not a single certain outcome. ## Switching to Distribution Thinking Instead of "Arsenal will score 2 goals," think: - P(0 goals) = 8% - P(1 goal) = 22% - P(2 goals) = 27% - P(3 goals) = 23% - P(4+ goals) = 20% This distribution (derived from a Poisson model with Arsenal's expected goals rate) is far more useful than a single point prediction. It allows you to calculate the probability of any total-goals outcome, any over/under line, and any scoreline. ## The Confidence Interval Discipline For any prediction, always estimate a range rather than a point: "I estimate Arsenal win probability at 52%, with a 90% confidence interval of 43–61%." The width of the confidence interval reflects genuine uncertainty. A narrow interval means you have high-quality information. A wide interval means the outcome is genuinely uncertain — which should affect stake size. ## Expected Value from a Distribution The full over/under model derives expected value from the entire probability distribution: EV(Over 2.5 at 1.91) = P(over 2.5) × 0.91 + P(under 2.5) × (−1) = (sum of P(3 goals) + P(4 goals) + ...) × 0.91 − (sum of P(0,1,2 goals)) × 1 Distribution thinking enables exact EV calculations rather than rough estimates.