Gamma, Exponential and Weibull Distributions explained for betting

In one sentence

The exponential, gamma and Weibull distributions describe waiting times: how long until the next event, or until several events have happened.

How it works

Poisson counts how many goals happen in 90 minutes. Flip the question round and ask how long until the first goal, and you get the exponential distribution. They are two views of the same process: a steady rate of events.

The exponential has a famous quirk called memorylessness. If a match is 0-0 at 30 minutes, the chance of a goal in the next 15 minutes is the same as it was in the first 15, assuming the rate has not changed. The clock does not "owe" you a goal.

The gamma distribution is the waiting time until the second, third or nth event, such as the time until two goals have been scored. The Weibull adds a shape setting that lets the rate rise or fall over time, which suits football, where goals come faster late in matches.

The maths

\text{Exponential: } P(T > t) = e^{-\lambda t}

\text{Gamma (time to the } n\text{th event): } P(T_n \le t) = P(N(t) \ge n), \quad N(t) \sim \text{Poisson}(\lambda t)

\text{Weibull: } P(T > t) = e^{-(t/s)^{k}}

T is the waiting time until the event.
λ is the event rate per unit of time (goals per minute).
n is how many events you are waiting for.
s is the Weibull scale (a typical waiting time) and k its shape: k above 1 means the rate rises over time, k below 1 means it falls, k = 1 is the exponential.

In words: at a steady rate, the chance of nothing happening decays smoothly with time; the Weibull lets that decay speed up or slow down.

Worked betting example

First Half Goals and Over/Under, illustrative inputs. A match has 2.6 expected goals, spread evenly, so the rate is 2.6 ÷ 90 ≈ 0.0289 goals per minute.

P(no goal in the first 30 minutes) = e to the power (−0.0289 × 30) = 42.04%. Fair odds 2.38.
The median time to the first goal is ln(2) ÷ 0.0289 ≈ 24.0 minutes.
P(at least one first-half goal) = 1 − e to the power (−1.3) = 72.75%. Fair odds 1.37.
Gamma for two goals: P(second goal arrives before 45 minutes) = 37.32%, the same as P(Poisson with mean 1.3 gives 2 or more). Fair odds for over 1.5 first-half goals ≈ 2.68.
In-play at 30 minutes, still 0-0: by memorylessness, P(goal in the next 15 minutes) = 1 − e to the power (−0.0289 × 15) ≈ 35.17%, ignoring stoppage time.

Real rates are not flat: scoring tends to rise as the match goes on, so a Weibull or time-varying rate usually prices the second half better.

Where it's good

In-play timing markets: next goal, goal before a set minute, first-half totals.
Trading around time decay in under/over and correct score markets.
Modelling gaps between price moves or matched bets in market data.
Second Half Goals: time to the first goal after the break, with the Weibull capturing the late rise.

Limitations and pitfalls

A constant rate is rarely true. Score, red cards, fatigue and stoppage time all change it.
Memorylessness is a model property, not a law of sport. Pressure can build, and markets often behave as if it does.
Stoppage time adds minutes that a naive 90-minute clock ignores, biasing late-goal prices.
Weibull shape is hard to estimate from small samples and easy to overfit.

How to build it

scipy.stats.expon, gamma and weibull_min in Python; lifelines for survival-style fitting.
Data: minute-by-minute event data (goal times, cards) with match state.
Tip: fit rates by match period and score state; the difference between minutes 0-15 and 75-90 is usually significant.

Poisson processes: the event stream these waiting times come from.
Poisson: counts over a fixed period.
Survival analysis: regression models for waiting times.
Negative binomial: where the gamma reappears as a mixing distribution.