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Probability Lab

See the outcomes before you count them.

Axiom-by-axiom proofs, Venn regions you can move, the full 36-cell dice grid, every coin sequence and every draw without replacement — each one naming its elementary outcomes and why they are equally likely.

Build the rules from the axioms

Problem 1 — every line labelled with the axiom it uses.

Axiom 1

P(E) ≥ 0 for every event E

Axiom 2

P(S) = 1

Axiom 3

Disjoint events add: P(∪Eᵢ) = ΣP(Eᵢ)

1P(∅) = 0

  1. S = S ∪ ∅ and S ∩ ∅ = ∅the two sets are disjoint
  2. P(S) = P(S) + P(∅)Axiom 3 — countable additivity on disjoint sets
  3. 1 = 1 + P(∅)Axiom 2 — P(S) = 1
  4. P(∅) = 0subtract; Axiom 1 keeps it ≥ 0

2Complement rule: P(Aᶜ) = 1 − P(A)

  1. S = A ∪ Aᶜ, A ∩ Aᶜ = ∅disjoint decomposition of S
  2. P(S) = P(A) + P(Aᶜ)Axiom 3
  3. 1 = P(A) + P(Aᶜ)Axiom 2
  4. P(Aᶜ) = 1 − P(A)rearrange

3Monotonicity: A ⊆ B ⇒ P(A) ≤ P(B)

  1. B = A ∪ (B \ A), A ∩ (B\A) = ∅disjoint because B\A excludes A
  2. P(B) = P(A) + P(B \ A)Axiom 3
  3. P(B \ A) ≥ 0Axiom 1 — non-negativity
  4. P(A) ≤ P(B)adding a non-negative number cannot decrease

40 ≤ P(A) ≤ 1

  1. P(A) ≥ 0Axiom 1
  2. A ⊆ Severy event lives in the sample space
  3. P(A) ≤ P(S) = 1monotonicity + Axiom 2

5Difference rule: A ⊆ B ⇒ P(B \ A) = P(B) − P(A)

  1. B = A ∪ (B \ A), disjointsame decomposition as above
  2. P(B) = P(A) + P(B \ A)Axiom 3
  3. P(B \ A) = P(B) − P(A)rearrange — valid only when A ⊆ B

Each rule uses only the three axioms and rules already derived above it — never cite a later line earlier.

Retrieval table

Problem 10 — cover this and write your own answers first.

P(Aᶜ) =

1 − P(A)

P(A ∪ B) =

P(A) + P(B) − P(A ∩ B)

If A ⊆ B, then

P(A) ≤ P(B) and P(B\A) = P(B) − P(A)

Equally likely finite outcomes give

P(E) = |E| / |S|

Ordered selection without repetition uses

P(n, r) = n!/(n−r)!

Unordered selection without repetition uses

C(n, r) = n!/(r!(n−r)!)

“Count first” is weaker than “define and justify the elementary outcomes first”, because a count is only meaningful once you know what you are counting and that those things carry equal probability.