Sample space: two observation scales for two coin tosses, and outcomes that are not equally likely

Goal and scope

Random experiment: toss a coin twice. (1) Recording the results in order, write the sample space S, and prove that it contains every possible elementary outcome and has exactly 4 elements; (2) recording only the number of heads, write the sample space S′={0,1,2}; (3) under a separately given probability model P (each ordered outcome in S has probability 1/4), compute the probabilities 1/4, 1/2, 1/4 of the outcomes in S′, showing that outcomes in one sample space need not be equally likely; (4) use another valid, non-uniform model Q on S to show that the sample space by itself does not fix probabilities; (5) distinguish the outcome (H,T) from the event {(H,T),(T,H)}.

Works inside the MKO's scope: the experiment and observation scale are explicit (assumption a1), and the sample space contains every possible elementary outcome under that description (assumption a2). Probability values are computed only in explicitly given models (G4, G5), as additional data of a later probability model, never derived from the sample space. It does not treat continuous or infinite sample spaces or measure theory, and does not change the canonical MKO or its English edition.

Givens and symbols

• cartesian_product_def
• finite_counting_basics
• prob_axioms
• image_preimage_def
• finite_prob_weights

• G1 — Experiment: toss the same coin twice. Each toss results in H (heads) or T (tails), with no other result (landing on edge is excluded) (assumption a1).
• G2 — Observation scale 1: record both results in order, as an ordered pair (first, second).
• G3 — Observation scale 2: record only the number of H among the two tosses.
• G4 — Probability model P (additional data, not derived from the sample space): each ordered outcome in S has probability 1/4, as used for example when the coin is fair and the tosses are independent.
• G5 — Another probability model Q (additional data): Q((H,H))=9/25, Q((H,T))=6/25, Q((T,H))=6/25, Q((T,T))=4/25, as used for example when H has probability 3/5 and the tosses are independent.

• H,\ T — heads, tails
• S — the sample space when both results are recorded in order
• S' — the sample space when only the number of heads is recorded
• h — the function sending an ordered outcome to its number of heads
• P,\ Q — two different probability models on S
• P' — the probability on S′ obtained from P through h

Rules and sources

• cartesian_product_def · prerequisite — Cartesian product: A×B is the set of ordered pairs (a,b) with a∈A and b∈B; (a,b)=(c,d) ⇔ a=c and b=d.
• finite_counting_basics · prerequisite — Finite counting: for disjoint finite sets A, B, |A∪B|=|A|+|B|; |A×B|=|A|·|B|; there is a bijection between finite sets ⇔ they have the same number of elements.
• prob_axioms · prerequisite — Probability axioms: P(A)≥0; P(Ω)=1; the probability of a (countable) union of pairwise disjoint events is the sum of their probabilities. Hence P(∅)=0 and P(Aᶜ)=1−P(A).
• image_preimage_def · prerequisite — Image and preimage: f(S)={f(a) ∣ a∈S}; f⁻¹(T)={a∈A ∣ f(a)∈T}. The preimage notation does not presuppose that f has an inverse.
• def_intro · prerequisite — Introducing a definition: a statement, expression or object may be given a name, after which the name and what it stands for are interchangeable.
• set_builder · prerequisite — Set-builder notation: x∈{y∈A ∣ P(y)} if and only if x∈A and P(x).
• case_analysis · prerequisite — Proof by cases: if P∨Q holds and both P⇒R and Q⇒R are proved, then R holds. The cases must cover every possibility.
• set_extensionality · prerequisite — Set equality: A=B if and only if A⊆B and B⊆A, i.e. they have the same elements.
• forall_intro · prerequisite — ∀-introduction (generalization): if a statement is proved for an arbitrary element x, with no extra assumption on x, it holds for every x in the domain.
• def_unfold · prerequisite — Unfolding a definition: replacing a defined term by its definition (or folding it back) does not change whether a statement is true.
• forall_elim · prerequisite — ∀-elimination (instantiation): if ∀x∈D, P(x) holds and a∈D, then P(a) holds.
• eq_replace · prerequisite — Replacement of equals: an expression inside a statement may be replaced by an equal expression without changing whether the statement is true.
• numeric_evaluation · prerequisite — Exact evaluation of an explicit integer or rational expression; independently recomputable.
• function_def · prerequisite — Function: f:A→B assigns to each a∈A exactly one f(a)∈B. Two functions are equal ⇔ they have the same domain and codomain and agree at every a.
• subset_def · prerequisite — Subset: A⊆B means every element of A is an element of B, i.e. ∀x (x∈A⇒x∈B).
• and_intro_elim · prerequisite — Conjunction: from P and Q infer P∧Q; from P∧Q infer P and infer Q.
• mko_sample_space · cited — If a random experiment has sample space S, every possible elementary outcome x satisfies x∈S. Membership in the space does not itself assume that outcomes are equally likely. · mko-sample-space
• finite_prob_weights · prerequisite — Point weights on a finite sample space: if each outcome ω is given p(ω)≥0 and all the p(ω) sum to 1, then setting P(E)=Σ_{ω∈E} p(ω) for every subset E of S gives a P that satisfies the probability axioms.
• L_S_complete · lemma — Under G1 and G2, every possible elementary outcome belongs to S={(H,H),(H,T),(T,H),(T,T)}.

Main derivation

\text{experiment: two tosses, recorded in order}

\text{outcome}=(a,b),\ a,b\in\{H,T\}

\{(a,b)\mid a\in\{H,T\},\ b\in\{H,T\}\}

\{H,T\}\times\{H,T\}

\{H,T\}\times\{H,T\}

S=\{(H,H),(H,T),(T,H),(T,T)\}

x\ \text{a possible elementary outcome}

x\in S

|\{H,T\}\times\{H,T\}|

|S|=2\cdot 2=4

\text{ordered outcome }(a,b)\in S

h:S\to S',\ h(a,b)=\text{number of }H\text{ among }a,b,\ S'=\{0,1,2\}

h(H,H),\ h(H,T),\ h(T,H),\ h(T,T)

2,\ 1,\ 1,\ 0;\quad h(S)=\{0,1,2\}=S'

|S|=4,\ |S'|=3

S\neq S'

p(\omega)=\tfrac14\ \ (\omega\in S)

P\ \text{is a probability on }S

h^{-1}(\{0\}),\ h^{-1}(\{1\}),\ h^{-1}(\{2\})

\{(T,T)\},\ \{(H,T),(T,H)\},\ \{(H,H)\}

P'(k)=P(h^{-1}(\{k\}))

P'(0)=\tfrac14,\ P'(1)=\tfrac12,\ P'(2)=\tfrac14

P'(1)=\tfrac12,\ P'(0)=\tfrac14

P'(1)\neq P'(0)

Q:\ \tfrac{9}{25},\ \tfrac{6}{25},\ \tfrac{6}{25},\ \tfrac{4}{25}

Q\ \text{is a probability on }S,\ Q\neq P

(H,T)\in S,\ E=\{(H,T),(T,H)\}

(H,T)\in E\subseteq S,\ (H,T)\neq E

\text{two observation scales and two models}

S=\{H,T\}^2,\ |S|=4;\ S'=\{0,1,2\};\ P'=(\tfrac14,\tfrac12,\tfrac14);\ Q\neq P

Substeps and lemmas

x\ \text{possible}

x=(a,b),\ a\in\{H,T\},\ b\in\{H,T\}

x=(a,b),\ a,b\in\{H,T\}

x\in S

Reverse audits

• S15 · sufficient_goal_reduction · S04, S05, S07, S08, S11, S12, S13, S14, and_intro_elim — Goal G is the conjunction of the five conclusions of the problem; each is proved forwards from the givens by the listed steps, and this step only forms the conjunction; G is never used as a premise.
• S14 · premise_audit · S03, S10, subset_def — E⊆S follows because both elements of E lie in S (a complete check). It is not claimed that outcomes and events are interchangeable.
• S13 · premise_audit · G5, finite_prob_weights, numeric_evaluation — Validity of Q needs only nonnegativity and total 1 (N6); the link to independence only explains where the weights come from, and the argument does not depend on it (N7 to N9 only check the arithmetic).
• S12 · premise_audit · S11, numeric_evaluation — Only two explicit rationals are compared (N5). The conclusion concerns the model G4; it is not claimed that S′ is non-uniform under every model.
• S11 · premise_audit · S09, S10, prob_axioms, finite_prob_weights — Additivity is used only for the disjoint single-point events {(H,T)} and {(T,H)}; the values of P′ depend entirely on the model G4, not on S′ itself.
• S10 · inverse_equivalence · image_preimage_def, set_extensionality — Each preimage is decided element by element over the four elements of S, so both sides have the same elements and the step holds both ways.
• S09 · premise_audit · G4, finite_prob_weights, prob_axioms — It needs the two conditions nonnegativity and total 1; N2 checks the latter. It does not claim that G4 is the only possible model (S13 gives another).
• S08 · premise_audit · S05, S07, set_extensionality — Different sizes are enough to show that the sets differ. Both are legitimate sample spaces, each relative to its own observation scale.
• S07 · premise_audit · S03, S04, image_preimage_def, L_S_complete — Since S contains every possible outcome (S04), h(S) is the set of all possible numbers of heads; this is a complete computation over the four elements of S, not a sample.
• S06 · premise_audit · G3, def_intro — This step only defines; completeness of S′ is checked in S07 through the values of h.
• S05 · premise_audit · finite_counting_basics, cartesian_product_def — It needs {H,T} to have two distinct elements (H≠T); the conclusion is a count and says nothing about probabilities.
• S04.2 · premise_audit · S02, S03, forall_intro — It needs the set equality of S03; it does not claim that the elements of S are equally likely.
• S04.1 · premise_audit · G1, S01 — Only the descriptions of the experiment and of the record in G1 and G2 are used; x is arbitrary.
• S04 · premise_audit · L_S_complete, G1 — Completeness rests on G1's 'each toss is H or T'; if the experiment allowed the coin to land on edge, S would not be complete.
• S03 · inverse_equivalence · case_analysis, cartesian_product_def, set_extensionality — The case split exhausts the two values of {H,T}, so the listed set and the product have the same elements: each listed pair is in the product, and each element of the product is listed.
• S02 · inverse_equivalence · cartesian_product_def — This directly rewrites the definition of a Cartesian product; both directions are given by the definition.
• S01 · premise_audit · G1, G2, mko_sample_space — This step only makes the experiment and observation scale explicit (assumption a1) and derives no new fact; a different way of recording gives a different sample space (see S06).

Error controls and negative controls

One sample space does not mean equally likely

The MKO names the mistake of believing that outcomes in one sample space must be equally likely. S′={0,1,2} is a legitimate sample space, yet under the model P its three outcomes have probabilities 1/4, 1/2, 1/4 (S11), not 1/3 each. Even the uniform weights on S are separate data from G4; the model Q of G5 is not uniform on the same S (S13).

An outcome is not an event

The MKO warns against confusing a single outcome with an event set. (H,T) is an element of S; 'exactly one head' is, in S, the event {(H,T),(T,H)} with two outcomes (S14). The same description corresponds to the outcome 1 in S′ but to an event in S; one must first say which sample space is in use.

Omitting a possible outcome

When results are recorded in order, (H,T) and (T,H) are different outcomes (S05). Writing S as {(H,H),(H,T),(T,T)} omits the possible outcome (T,H) and violates assumption a2; later event and probability computations would lose their correct domain. The proof of L_S_complete (S04.1, S04.2) is what rules out such an omission.

• S′={0,1,2} is a sample space, so each outcome has probability 1/3 → reject — Under the model P, P′(1)=1/2≠1/3 (S11); a sample space carries no assumption of equal likelihood.
• S has 4 elements, so (H,H) must have probability 1/4 → reject — On the same S, the model Q gives (H,H) probability 9/25 (S13).
• When results are recorded in order, {(H,H),(H,T),(T,T)} is the sample space → reject — (T,H) is a possible outcome but is missing, violating assumption a2.
• (H,T) and {(H,T),(T,H)} are the same object → reject — The first is an ordered pair (an outcome), the second a two-element set (an event).
• S and S′ differ, so at least one of them is not a sample space of this experiment → reject — Each is relative to a different observation scale (G2, G3), and both are legitimate sample spaces (S04, S07).

Result and limitations

Recording two coin tosses in order, the sample space S={(H,H),(H,T),(T,H),(T,T)} contains every possible elementary outcome and has exactly 4 elements; recording only the number of heads gives the sample space S′={0,1,2}. Under the separately given model P, the outcomes of S′ have probabilities 1/4, 1/2, 1/4 and are not equally likely; S also carries the non-uniform model Q. So a sample space only says which outcomes are possible and does not fix probabilities. (H,T) is an outcome; {(H,T),(T,H)} is an event.

• G4 and G5 are modelling assumptions; this guide does not discuss whether they fit an actual coin, which is a statistical question.
• finite_prob_weights is used as a prerequisite and is not derived from the probability axioms in this guide.

Verification and source binding

11 declared checks for this guide; only the listed expressions are covered.

• source commit: b28a5edbd46279b1db2a3f3f48bb3f447e18899f
• adopted source SHA-256: c0d1f9b0e3d3e27afde0183074318b955b8122887189e0f0232556ac536804a1
• original source SHA-256: c0d1f9b0e3d3e27afde0183074318b955b8122887189e0f0232556ac536804a1
• Lean: NOT_RUN · independent AI: NOT_MEASURED

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