Mathematics · Introduction to Probability and Combinatorics · Fundamental Counting and Probability Models
Outcomes, Events, & Models
Last edited 9/30/2026, 4:33:14 AM
One Experiment, Two Records
Two distinguishable dice are rolled. One observer records both faces, while another records only their sum. The first observer can record (1,6) and (3,4) separately. The second writes 7 in either case.
There are eleven possible sums, from 2 through 12. Does that make the chance of a sum of 7 equal to 1/11?
Tempting, right? Eleven sums, so 1/11 each. But that's smuggling in an assumption — that the eleven sums are the equally-likely things, when nobody said that. What's actually equally likely here?
Models of Random Experiments
Definition
Sample Space
A sample spaceΩ is the set of outcomes in a mathematical model. An event is an allowed subset of Ω: the event occurs when the realized outcome belongs to that subset. For a finite sample space, we allow every subset as an event. For real-valued outcomes, we will use intervals and other measurable sets; Lesson 2 explains why an event collection is part of the model.
The sampling protocol defines how the outcome is selected. Listing Ω alone does not give its probabilities.
For example, Ω={a,b,c} can describe a uniformly selected label, or a device that selects a half the time and each other label one quarter of the time. Both have the same possible records, however, their probability assignments differ.
Think of total probability as one dollar you're handing out across all the outcomes. w(ω) is just how much each outcome gets. This is what we mean when we said the protocol tells you how the outcome gets picked. The weights are the protocol, written as numbers.
Definition
Finite Weighted Models
In a finite weighted model, assign numbers w(ω)≥0 whose sum is 1. Define
P(A)=ω∈A∑w(ω).
This is a model definition. Lesson 2 will show how it fits the general probability axioms. A finite uniform model assigns the same weight to every outcome. If there are N>0 outcomes and their common weight is c, normalization gives Nc=1, so c=1/N. Consequently,
P(A)=N∣A∣.
The equal-weight assumption does the essential work. Counting favorable records is valid only after that assumption has been justified or specified.
Grouping outcomes without changing the experiment
Suppose g:Ω→S describes the information we keep. The event that the new record equals s consists of all outcomes mapped to s:
{ω∈Ω:g(ω)=s}.
Theorem (Probability of a Grouped Record)
In a finite weighted model,
P(g=s)=ω:g(ω)=s∑w(ω).
Proof
All we're doing is pushing several little piles of weight into one bigger pile labeled s — the piles don't overlap, so the total is just the sum. (This is exactly what happens with the six dice pairs landing on sum 7 below.) The outcomes on the right are exactly those making the recorded statement true. Applying the definition of event probability to that subset gives the formula. No new random selection has occurred. We have only grouped the original weights. In a uniform model the formula becomes the number of outcomes in the group divided by ∣Ω∣. Thus equally likely grouped records require equal group sizes. This last conclusion concerns a uniform starting model; for unequal weights, total group weight is what matters.
∎
This is useful whenever a problem records a sum, a color, a winner, or some other summary instead of the underlying outcome.
Resolve the dice question
Worked example PC-M01-01-W01
Use the model in which the 36 ordered face pairs are equally likely. This specifies the joint model, not merely the behavior of each die by itself. The pairs with sum 7 are
(1,6),(2,5),(3,4),(4,3),(5,2),(6,1).
There are six, so P(sum 7)=6/36=1/6. Only (1,1) has sum 2, so its probability is 1/36. The eleven sum records are not equally likely. As a quick gut check, if you added up all eleven sum-probabilities this way, they'd total exactly 1 — no weight lost, none double-counted.
Why keep order when addition ignores it? The dice are distinguishable, and (1,6) and (6,1) are different equally weighted outcomes. Adding their weights is how we recover the probability of the summary.
Two further models
Worked example PC-M01-01-W02: visible labels
A container has four distinguishable tickets. Three display a triangle and one displays a circle. Choose a ticket uniformly. The four physical tickets are equally likely; the two displayed shapes are not. The triangle event contains three ticket outcomes, so its probability is 3/4.
If instead a program first chooses one of the two shapes with equal probability and then shows that shape, the triangle probability is 1/2. These are different protocols, not contradictory answers.
Worked example PC-M01-01-W03: a location on a segment
A device selects a location on a segment of length 3. Here's the snag: there's no "number of points" in a segment to divide by, so counting is out. We need a different notion of size. Specify the uniform-length model by the rule
P(location in [a,b])=3b−a,0≤a≤b≤3.
This is the model we choose, not something implied by the word “random.” The first unit has probability 1/3, while the interval from 1 to 3 has probability 2/3. A rule that instead favors locations near the right end would assign different probabilities. Lesson 6 constructs interval probabilities from integrals and explains the role of density.
The move from counting to length changes the way weights are assigned, while the question “which outcomes belong to the event?” remains the same.
What information is enough?
A coarse record can be convenient if it preserves the question. A sum is enough to decide whether two dice total 7. It is not enough to decide whether the first die exceeds the second: both (1,6) and (6,1) produce the same sum but give different answers.
Before choosing a sample space, state the experiment, the recorded outcome, the selection rule, and the event. If a probability cannot be determined, identify the missing rule instead of assuming uniformity.
Practice begins with identifying events, then grouping weights, then comparing protocols. Attempt the exercises before using the separate hints or solutions.
Sources
Original exposition using the finite-model framework in RO, §§2.1–2.3, and GS-local, §1.2. The segment model is developed further using GS-local §2.2 in Lesson 6. No external exercise statement is reproduced.
