Lesson 5 of 11 · 7 min
Axioms of probability
NCERT §14.2
The stall team argues about the coin. One says heads and tails must each get 1/2. Another says a bent coin could be 1/4 and 3/4. A third writes 0.7 and 0.7. Which assignments are even allowed?
The lesson in notes
In short
The axiomatic approach does not say how a number is found; it lays down the rules every valid assignment of probabilities must obey.
P is a real-valued function defined on the events of S (its power set), taking values in [0, 1], that obeys three rules: (i) every event E has P(E) ≥ 0; (ii) the sure event has probability 1, P(S) = 1; (iii) for two mutually exclusive events E and F, the probability of their union is P(E) + P(F).
Consequence: P(φ) = 0. Put F = φ in (iii); E and φ are disjoint and E ∪ φ = E, so P(E) = P(E) + P(φ).
For S = {ω₁, …, ωₙ}: every 0 ≤ P(ωᵢ) ≤ 1, the P(ωᵢ) add to 1, and for any event A, P(A) is the sum of P(ωᵢ) over the ωᵢ in A. The singleton {ωᵢ} is an elementary event, and P(ωᵢ) is shorthand for P({ωᵢ}).
A coin may be given P(H) = 1/2, P(T) = 1/2, or equally validly P(H) = 1/4, P(T) = 3/4. Any p and 1 − p with 0 ≤ p ≤ 1 satisfies the axioms, so there are infinitely many valid assignments.
NCERT's Example 4 on six outcomes: an assignment of 1/6 each is valid; 1, 0, 0, 0, 0, 0 is valid; one with two negative values is not; one with a value 3/2 is not; and 0.1, 0.2, …, 0.6 is not, because the sum is 2.1.
Checklist for any table of probabilities: no entry below 0, no entry above 1, and the entries add to exactly 1.