ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-08-13

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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A loaded die as a nonuniform finite probability space

Example

On $Ω = {1, 2, 3, 4, 5, 6}$ assign weights $w (1) = 0, w (2) = \frac{1}{10}, w (3) = w (4) = \frac{1}{5}, w (5) = w (6) = \frac{1}{4} .$ This is a loaded die with a possible zero-weight outcome. Let $A = {4, 5, 6}$ and $B = {2, 4, 6}$ .

Facts & Assumptions

Given: The weights and events in the Example.

[L1]

Finite probability spaces allow nonnegative outcome weights summing to $1$ (Finite probability spaces, outcome weights, events, and event probabilities).

[L2]

Complements and differences obey the finite probability laws (Normalization, nonnegativity, monotonicity, complements, and differences in a finite probability space).

[L3]

$P (A ∣ B) = P (A \cap B) / P (B)$ when $P (B) > 0$ (Conditional probability $P (A ∣ B)$ for $P (B) > 0$ ).

[L4]

The multiplication rule is $P (A \cap B) = P (A ∣ B) P (B)$ (The multiplication rule and finite chain rule for conditional probability).

Verification

technique · direct

1.1

The six weights sum to $1$ , so [L1] gives a finite probability space; ${1}$ is nonempty but has probability zero.

L1algebra

1.2

Direct summation gives $P (A) = 7 / 10$ , $P (B) = 11 / 20$ , and $P (A \cap B) = 9 / 20$ .

L1L2algebra

2.1

Since $P (B) > 0$ , [L3] gives $P (A ∣ B) = (9 / 20) / (11 / 20) = 9 / 11$ .

step 1.2L3algebra

3.1

Finally $(9 / 11) (11 / 20) = 9 / 20 = P (A \cap B)$ , verifying [L4] in this nonuniform space.

step 1.2step 2.1L4algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Section 1.2 (standard reference, not scraped)