ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 39 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)