Probability Essentials
Author: Vitaly - mr. Koteo (Brisbane Mafia Club)
Probability Essentials
Previous chapter explained why Sports Mafia can be viewed as a mathematical game: the game starts with many possible worlds and gradually eliminates them under structural constraints.
This chapter introduces the probability layer that sits on top of that framework.
We are no longer asking only:
“Which worlds are possible?”
We now ask:
“Given what has happened so far, which of the possible worlds are more likely?”
Probability in Mafia is not about exact numbers. It is about updating beliefs consistently as new information appears.
From Combinatorics to Probability
In Previous Chapter we used the function:
C(N, D)to describe how many Dark-team configurations are still possible, given:
Nplayers aliveDDark players alive
This is a combinatorial view of the game.
In this chapter, we do not replace that framework. We build on it.
Key distinction:
C(N, D)defines the space of possible worldsProbability assigns weights to those worlds based on observed events
All probability reasoning in this chapter operates inside the space defined by C(N, D).
Basic Probabilities: Roles, Checks, and Baselines
At the start of the game:
10 players are alive
3 are Dark
7 are Red
Before any information is revealed, the baseline probability that a randomly chosen player is Dark is:
This is a prior probability — a starting point, not a conclusion.
After the first night kill (which is almost always Red in standard play):
9 players remain alive
3 are Dark
The new baseline becomes:
This change is mechanical. It does not reflect skill, reads, or insight — only the reduced player pool.
Sheriff Checks: What the Numbers Actually Mean
Suppose it is early in the game, after Night 1:
9 players alive
3 Dark players alive
Before any check is revealed, and assuming no targeting bias, the baseline probabilities for a Sheriff check are:
This does not mean:
the Sheriff checks randomly in practice
Red checks are “weak”
checks are independent forever
It means something precise and limited:
Before information is revealed, the expected distribution of colors among unchecked players reflects the underlying proportions.
Why this matters:
Early Red checks are expected, not surprising
Early Red checks are still valuable, because they reduce uncertainty
The mistake is not valuing checks — the mistake is treating early information as final certainty
Any information is better than no information. Probability teaches how much weight to give it.
Conditional Probability (Bayesian Thinking Without the Pain)
Probability in Mafia is always conditional.
Instead of asking:
“Is player #6 Dark?”
The correct question is:
“How likely is player #6 Dark given everything that has already happened?”
This is written as:
Every new event updates probabilities:
a check
a vote
a kill
a reveal
a contradiction
The core Bayesian idea can be written compactly as:
You rarely need exact numbers. What matters is:
which possibilities become more likely
which become less likely
which become impossible
What Kills Mean Mathematically
A night kill is not a verdict. It is a strategic state change.
From a mathematical perspective, a kill:
removes a player from all remaining possible worlds
removes that player’s future votes and speeches
accelerates the approach to parity
increases pressure on the Red team by reducing margin for error
A kill does not:
prove alignment
erase information already produced
invalidate past votes or checks
Kills are best interpreted as:
the Dark team removing players dangerous to its strategy and timeline.
Kills constrain future probability evolution, not past information.
Sheriff Probability Trees
When one or more players claim the Sheriff role, the game branches into multiple competing sets of possible worlds.
From the Red team’s perspective, each Sheriff claim creates a hypothesis:
Worlds where this Sheriff is real
Worlds where this Sheriff is fake
If two Sheriffs reveal, the table is no longer evaluating personalities — it is comparing two incompatible models of reality.
Checks as Mathematical Constraints
A Sheriff check — whether real or fake — does not immediately certify truth. It restricts the set of possible worlds.
Each claimed check imposes a condition:
the checked player must be Red in worlds where the Sheriff is real
the checked player remains unconstrained in worlds where the Sheriff is fake
As more checks are revealed, some worlds become mathematically impossible because they violate hard constraints such as:
the total number of Dark players remaining
the number of players still alive
already confirmed Dark vote-outs
parity conditions (game continuation vs immediate Dark win)
A Sheriff line becomes invalid not because it “sounds wrong”, but because no Dark-team configuration exists that satisfies all constraints simultaneously.
Why Fake Sheriffs Eventually Collapse
A fake Sheriff may claim:
Red checks,
include Dark teammates as “Red”,
and remain consistent for several rounds.
However, fake claims must still obey:
player-count arithmetic,
remaining Dark limits,
and endgame parity rules.
As the game progresses and N decreases, the number of Dark players that can still be hidden shrinks.
At some point, a fake Sheriff line requires more Red players than the game state allows, or implicitly forces the Dark count below zero.
When no valid Dark-team configuration exists under a claimed check sequence, that entire branch is eliminated.
This is how probability trees are pruned — not by belief, but by arithmetic.
Key Mathematical Insight
Sheriff checks do not “prove” alignment directly. They progressively eliminate impossible worlds until only compatible ones remain.
This is why even fake information still accelerates convergence: it limits future claims and increases the chance of contradiction.
Expected Value (EV) of Decisions
Probability tells us how likely something is. Expected Value (EV) tells us how good a decision is, on average.
These are not the same.
What EV Means in Mafia
Expected Value is the weighted average outcome of a decision across all possible worlds still considered plausible.
Formally:
In Sports Mafia, outcomes are not numbers — they are game states:
Red advantage vs Dark advantage
survival vs parity loss
information gain vs information collapse
A decision’s EV depends on:
how many possible worlds it improves,
how badly it damages the remaining ones,
and how irreversible the downside is.
Correct vs Incorrect Outcomes
A move can be:
correct in EV but fail due to unlucky world selection
incorrect in EV but succeed accidentally
This distinction is crucial.
Example:
A Sheriff reveal that loses the game immediately is still correct if, across all plausible worlds, it had higher survival probability than delaying.
A “safe” vote that preserves ambiguity can be incorrect if it allows Dark parity in most remaining worlds.
EV judges decisions before outcomes are known.
Why EV Matters More Than Being Right
Sports Mafia is not won by guessing the correct world early. It is won by maximising Red survival across many plausible worlds.
Good players do not ask:
“Was I right?”
They ask:
“Given what we knew, was this the best decision?”
That mindset is what separates:
logical play from emotional play
strategic consistency from hindsight bias
Probability explains what is likely. EV explains what should be done.
Last updated