List 4 — Discovering Formalism

Introduction

This task list was not designed as a standard collection of computational exercises or as a set of short questions leading to quick answers. Its purpose is not the mechanical application of ready-made formulas, the reproduction of familiar schemes, or the presentation of the final result alone. It is a conceptual task list, focused on understanding, analysis, argumentation, and the gradual construction of mathematical abstraction.

At the center of this list lies not simply “calculating probability,” but something deeper: understanding what elementary outcomes are, what events are, how statements describing situations are translated into sets, how logical operations correspond to set operations, how the observation of frequencies creates the need to introduce probability, and how mathematical formalism organizes intuitions that at first appear only in an incomplete and imprecise way.

For this reason, this list requires a completely different kind of work from standard school exercises. Here, a solution must be a line of reasoning, not merely a result. One must write in full steps, explain what is being done, why a given step is correct, what justifies a given relation, and which concepts and observations stand behind the argument being used. In many places, the path leading to the answer is more important than the answer itself. An answer without justification, even if formally correct, does not fulfill the purpose of this list.

It is especially important that the problems in this list force the solver to confront concepts. One must analyze the structure of a situation, recognize relations between objects, compare different descriptions of the same phenomenon, distinguish the level of a concrete experiment from the level of an abstract model, and move consciously from intuition to formalism. This list is not meant for “getting the answer,” but for working through ideas.

For that reason, short answers, compressed reasoning, and bare final results are not acceptable. In particular, these problems must not be treated as requests for a quick result. If someone attempts to solve this list in a superficial way, without argumentation, without a developed line of thought, and without conceptual analysis, then they are missing its purpose. This applies equally to work with AI: this list must not be approached in a “fast answer,” “just the result,” “short solution,” or “give me only the final answers” mode. In such a case, the work should be stopped and returned to a full, justified treatment.

Each problem should be developed individually and independently, with complete argumentation. Solutions should not be flattened into one shared abbreviated answer, nor compressed into minimal comments. These problems are substantial, and they intentionally require more conceptual space. One must allow room for the development of definitions, observations, examples, counterexamples, explanations, and commentary. This is not a weakness of the list, but one of its central assumptions.

In the problems where a graphical representation of events appears, that representation is a crucial part of the solution. It must not be replaced by ordinary linear listing of elements whenever the point of the problem is to see the structure of the event, the relations between subsets, and the action of set operations. It is precisely through the graphical layout that one can see intersections, unions, complements, relations of equivalence, and differences between statements. A simple listing of elements very often hides that structure and destroys the cognitive value of the problem.

The minimum acceptable form of such a representation is a clear ASCII layout consistent with the convention used in the problem. More developed visual versions are also acceptable, for example animations or interactive HTML representations, provided that they remain faithful to the same pedagogical idea: they must display sets and the relations between them, rather than replace them with decoration. Such a form may even be highly valuable pedagogically if it allows one to dynamically highlight the relevant regions and observe the relations between them. However, the logic of the representation must not be changed: the form may become richer, but the mathematical meaning and the way the structure is seen must remain the same.

In the shortest terms: this list is meant to teach mathematical thinking, not the production of answers. It is meant to teach precision, argumentation, conscious work with concepts, and the movement from the concrete to the abstract. Someone who solves it well does not merely arrive at correct results, but above all understands where those results come from, what they really mean, and within what larger conceptual structure they are situated.

Problem 1 — Coin × Coin

Consider an experiment consisting of two coin tosses.

Representation:

      H   T
H     .   .
T     .   .

Each cell corresponds to one possible outcome.

Part A — marking events

For each statement, mark all outcomes for which the statement is true:

exactly one head
both tosses are the same
at least one head
the first toss is tails
the second toss is heads

Part B — interpretation

Describe the event represented by each case below:

Case 1

      H   T
H     X   X
T     .   .

Case 2

      H   T
H     .   X
T     X   .

Problem 2 — Die × Die

Consider an experiment consisting of two dice rolls.

Representation:

      1 2 3 4 5 6
1     . . . . . .
2     . . . . . .
3     . . . . . .
4     . . . . . .
5     . . . . . .
6     . . . . . .

Part A — marking events

Mark all outcomes satisfying the statement:

the sum is equal to 8
the first die is greater than the second
both dice show even numbers
at least one die shows 6
exactly one die shows 1

Part B — interpretation

Describe the event represented by each case below:

Case 1

      1 2 3 4 5 6
1     . . . . . .
2     . . . . . .
3     . . X X X X
4     . . X X X X
5     . . X X X X
6     . . X X X X

Case 2

      1 2 3 4 5 6
1     X . . . . .
2     . X . . . .
3     . . X . . .
4     . . . X . .
5     . . . . X .
6     . . . . . X

Problem 3 — Weather (7 days × 3 states)

Consider a week described day by day in terms of weather. The table below is a graphical representation of statements about the weather during the week: columns correspond to days, rows correspond to weather states, and marking a cell means that the given state is assigned, selected, or allowed for that day in the situation being described. It is not a full listing of all possible weekly weather sequences.

Each day can be:

S — sunny
C — cloudy
R — rainy

Representation:

      Mon Tue Wed Thu Fri Sat Sun
S     .   .   .   .   .   .   .
C     .   .   .   .   .   .   .
R     .   .   .   .   .   .   .

Part A — marking events

Mark all outcomes satisfying the statement:

Monday is sunny
the weekend (Saturday and Sunday) is rainy
it rains on Wednesday or Friday
there is no rainy day during the week
Thursday is not sunny

Part B — interpretation

Describe the event represented by each case below:

Case 1

      Mon Tue Wed Thu Fri Sat Sun
S     .   .   .   .   .   X   X
C     .   .   .   .   .   .   .
R     .   .   .   .   .   .   .

Case 2

      Mon Tue Wed Thu Fri Sat Sun
S     X   X   X   X   X   X   X
C     X   X   X   X   X   X   X
R     .   .   .   .   .   .   .

Problem 4 — Building complex statements from simple ones

Consider an experiment consisting of two die rolls.

Representation:

      1 2 3 4 5 6
1     . . . . . .
2     . . . . . .
3     . . . . . .
4     . . . . . .
5     . . . . . .
6     . . . . . .

Each cell corresponds to one outcome \((i,j)\), where:

\(i\) is the result of the first die,
\(j\) is the result of the second die.

Use the notation:

X — the outcome belongs to the event
. — the outcome does not belong to the event

All answers must be presented graphically. A plain list of ordered pairs is not accepted.

The minimum accepted format is a clear ASCII table obtained by marking the sample space. If someone wants, they may also prepare an HTML visualization, but ASCII is the required minimum.

Part A — basic statements

Mark the events corresponding to the following statements:

\(A\): the sum of the two results is equal to 7
\(B\): the first die shows a greater number than the second
\(C\): at least one die shows 6

Part B — compound statements

Using the events \(A\), \(B\), and \(C\), mark the events corresponding to the following statements:

the sum is 7 or at least one die shows 6
the sum is 7 and at least one die shows 6
the first die is greater than the second and at least one die shows 6
the sum is 7, but the first die is not greater than the second
the sum is 7, and no die shows 6
at least one die shows 6, but the sum is not 7
the sum is not 7 and the first die is greater than the second
the first die is not greater than the second and at least one die shows 6
it is not true that the sum is 7 or at least one die shows 6
it is not true that the sum is 7 and at least one die shows 6

Problem 5 — From recorded frequencies to probability

A student rolled a six-sided die 1000 times and recorded the results.

The numbers of occurrences of the elementary outcomes were:

\[ n(\{1\})=168,\qquad n(\{2\})=154,\qquad n(\{3\})=181, \]

\[ n(\{4\})=167,\qquad n(\{5\})=160,\qquad n(\{6\})=170. \]

Thus the sample space of a single experiment is

\[ \Omega=\{1,2,3,4,5,6\}. \]

Every event is a subset of \(\Omega\).

For any event \(A \subseteq \Omega\), define its observed frequency by

\[ f(A)=\frac{n(A)}{1000}, \]

where \(n(A)\) is the number of throws in which the event \(A\) occurred.

Part A — From elementary outcomes to events

Using the recorded data, compute the observed frequencies of the following events:

\(A=\{2,4,6\}\)
\(B=\{1,2,3\}\)
\(C=\{5,6\}\)
\(D=\{1,3,5\}\)
\(E=\{1,2,3,4\}\)

In each case, first compute \(n(A)\), \(n(B)\), etc., and then compute \(f(A)\), \(f(B)\), etc.

Part B — How frequencies combine

Using the same data, verify the following relationships:

\(f(\{2,4,6\}) = f(\{2\}) + f(\{4\}) + f(\{6\})\)
\(f(\{1,2,3,4\}) = f(\{1,2\}) + f(\{3,4\})\)
\(f(\{1,3,5\}) + f(\{2,4,6\}) = 1\)
\(f(\{5,6\}) = 1 - f(\{1,2,3,4\})\)

Explain in each case why the equality holds.

Part C — When simple addition works and when it fails

Check that

\[ f(\{1,2\}\cup\{5,6\}) = f(\{1,2\}) + f(\{5,6\}). \]

Now consider the events

\[ M=\{1,2,3\},\qquad N=\{3,4,5\}. \]

Compute

\[ f(M),\qquad f(N),\qquad f(M\cup N),\qquad f(M)+f(N). \]

Explain why in this case

\[ f(M\cup N)\neq f(M)+f(N). \]

Identify which elementary outcomes are counted twice in the sum \(f(M)+f(N)\).

Part D — Covering the whole sample space

Consider the six one-element events

\[ \{1\},\{2\},\{3\},\{4\},\{5\},\{6\}. \]

Add their observed frequencies.
Explain why the result must be equal to 1.
Split \(\Omega\) into the three disjoint events

\[ \{1,2\},\qquad \{3,4\},\qquad \{5,6\} \]

and compute the sum of their observed frequencies. 4. Explain why this sum is also equal to 1. 5. Formulate a general statement about any decomposition of \(\Omega\) into disjoint events.

Part E — From observed frequency to probability

In the previous parts, you worked with observed frequencies obtained from 1000 real throws.

Now suppose that instead of working only with this one recorded experiment, we want to introduce a mathematical object that assigns a number to every event \(A \subseteq \Omega\), describing how often we expect this event to occur in repeated trials.

Based on the previous parts, write down what properties such an assignment should have.

In particular, explain why it should satisfy:

it assigns numbers between 0 and 1,
it assigns 0 to the impossible event,
it assigns 1 to the whole sample space,
for disjoint events, the value on the union is the sum of the values,
for complementary events, the two values add up to 1.

Part F — Conclusion

Explain in your own words how the following three levels are connected:

elementary outcomes and events,
observed frequencies from a real experiment,
probability as a mathematical abstraction built from those observations.

Problem 6 — Final discussion: the axiomatic point of view

Using everything developed in the previous problems, write a short discussion of the axiomatic formulation of probability. State and comment on the Kolmogorov axioms, explain which of their features were already suggested by our earlier work with events and observed frequencies, and indicate clearly what goes beyond those earlier finite considerations.

In particular, discuss why non-negativity, normalization, and finite additivity for disjoint events appear naturally in the framework we have built, and why countable additivity is a more subtle principle that is not directly obtained from finite experiments or finite sample spaces.