No. 08February 2026.
Probability and statistics, MT2204: notes and an exercise
Probability from three axioms, then run forward in time
The course builds probability on three axioms: P(A) ≥ 0, P(Ω) = 1, and additivity over disjoint events. My notes prove six rules from them. Every proof makes the same move: split a set into disjoint pieces, then add.
| 1 | P(∅) = 0 | ∅, ∅, ∅, … are disjoint and their union is ∅, so x = x + x + …, and only x = 0 works. |
| 2 | P(Ac) = 1 − P(A) | A and its complement are disjoint and together make Ω. |
| 3 | A ⊆ B ⇒ P(A) ≤ P(B) | B = A ∪ (B ∖ A), two disjoint pieces, and P(B ∖ A) ≥ 0. |
| 4 | P(A ∪ B) = P(A) + P(B) − P(A ∩ B) | A ∪ B is three disjoint pieces, and P(A) + P(B) counts the overlap twice. |
| 5 | A1 ⊆ A2 ⇒ P(A2 − A1) = P(A2) − P(A1) | A₂ = A₁ ∪ (A₂ − A₁), then additivity. |
| 6 | 0 ≤ P(A) ≤ 1 | Nonnegativity below; the third rule with B = Ω above. |
Then an exercise put the rules to work. Alice is up to date or behind at the end of each week. Up to date, she stays so with probability 0.8; behind, she catches up with probability 0.6. She starts up to date. The total probability theorem gives each week from the one before, and rule 2 removes P(Bn):
P(Un+1) = 0.8 P(Un) + 0.6 P(Bn)=0.6 + 0.2 P(Un)
Week by week that is 0.8, 0.76, 0.752. My notes stop partway through week two; the figure carries on. The gap to ¾ shrinks fivefold each week, so P(Un) = ¾ + ¼ (0.2)ⁿ, and within a month it hardly matters where she started.
what I don’t understand yet
Bayes’ rule. My notes set two exercises, marbles drawn from one of two boxes and a watch found in a drawer, and solve neither yet. They are the next page.
where this goes
Entry 09 spends probabilities as widths: arithmetic coding gives each symbol a slice of [0, 1) as wide as its probability.