Probability | Class 11 Mathematics Notes

14.2 Axiomatic Approach to Probability

This section introduces the axiomatic approach to probability, which provides a rigorous mathematical foundation for assigning probabilities to events. Unlike classical or empirical approaches, the axiomatic method defines probability as a function P from the power set of the sample space S to the real interval [0,1], satisfying three axioms:

(i) Non-negativity: For any event E, P(E) ≥ 0.

(ii) Normalization: The probability of the entire sample space is 1, i.e., P(S) = 1.

(iii) Additivity: For any two mutually exclusive events E and F, P(E ∪ F) = P(E) + P(F).

From these axioms, it follows that the probability of the empty set φ is zero. The section further explains that for a finite sample space S = {ω₁, ω₂, ..., ωₙ}, the probability assigned to each outcome ωᵢ satisfies 0 ≤ P(ωᵢ) ≤ 1 and the sum of probabilities over all outcomes is 1. The probability of any event A is the sum of probabilities of outcomes in A. This approach allows flexible probability assignments, including unequal probabilities, as long as the axioms are satisfied. Examples illustrate valid and invalid probability assignments to outcomes.

📊 Diagram: Table on page 9 (6×7) showing various assignments of probabilities to outcomes and their validity.

🧪 Activity: Check validity of different probability assignments to sample space outcomes using axioms.

🔗 Connection: This section leads to calculating probabilities of events using the axiomatic approach and considering equally likely outcomes.

Table on page 9 (6×7)