Bayesian House Edge

Frequentist p-values vs. Bayesian posterior probability

Standard statistical audits use Frequentist hypothesis testing. They start by assuming the casino is honest (the null hypothesis) and calculate a p-value—the probability of experiencing your losses by pure luck. If the p-value is 0.02, they reject the hypothesis, indicating the game is rigged.

However, frequentist tests do not answer the question you actually care about: **what is the actual probability that the casino’s true house edge is higher than their claim?** To answer this, we must use Bayesian statistics.

By combining your “prior belief” (the casino’s advertised house edge and your confidence in it) with your “empirical data” (your session wagers, outcomes, and volatility), a Bayesian conjugate update calculates the exact probability distribution of the true house edge, identifying the true mathematical expectation (our guide to Expected Value Explained outlines why this matters).

The math: Normal-Normal conjugate update

Because your session results ($X$) converge to a normal distribution over a large number of trials due to the Central Limit Theorem, we can model both the prior and the data using normal distributions:

1. Defining the Prior (Claimed Edge)

We represent your prior belief about the house edge as a normal distribution:

Prior = N(μ_prior, σ²_prior)

Where $mu_{prior}$ is the claimed house edge, and $sigma^2_{prior}$ represents your uncertainty (a narrow variance means you trust the casino; a wide variance means you are highly skeptical).

2. Calculating the Data Likelihood

Your empirical session results yield an observed house edge ($mu_{data}$) with an associated variance ($sigma^2_{data}$):

μ_data = -1 * Net_Profit / Total_Wagered
σ²_data = Volatility² / Total_Wagered

3. The Posterior Conjugate Update

The posterior distribution of the true house edge is also normal, identifying the true mathematical expectation (our guide to Expected Value Explained outlines why this matters), $N(mu_{post}, sigma^2_{post})$, calculated as:

1 / σ²_post = 1 / σ²_prior + 1 / σ²_data
μ_post = σ²_post * (μ_prior / σ²_prior + μ_data / σ²_data)

The tool then integrates the resulting normal curve to find the exact probability that the true house edge is more than double the casino’s advertised claim:

P(True_Edge > 2 * Claimed_Edge) = 1 - Φ((2 * Claimed_Edge - μ_post) / σ_post)

Step-by-step audit: Auditing a slot claim

Let’s audit an online slot that claims a **2.00% house edge (98% RTP)** (see the difference in our guide on RTP vs House Edge). You play 10,000 spins with an average bet of $2 ($20,000 total volume). The slot has a standard deviation of 3.0. You end your session down $900, which represents an observed loss rate of 4.50%.

1. Enter the claimed edge (2.00%) and your prior confidence (e.g., standard prior variance).
2. Input your session data: total wagers ($20,000), net result (-$900), and game volatility (3.0).
3. Click “Verify.” The auditor will run the Normal-Normal conjugate formulas.
4. Analyze the posterior curve. If your observed loss is high and your volume is large, the posterior mean will shift toward 3.50%.
5. The tool will output the probability (e.g., 78%) that the slot’s true house edge is higher than advertised, giving you a clear, mathematical metric of potential operator manipulation.

Frequently asked questions