Question

Prove that $p(A)+p(\neg \mathrm{A})=1$ in the frequency interpretation (without using Kolmogorov's axioms).

   Prove that $p(A)+p(\neg \mathrm{A})=1$ in the frequency interpretation (without using Kolmogorov's axioms).

An Introduction to Decision Theory

An Introduction to Decision Theory

Martin Peterson 1st Edition

Chapter 7, Problem 3

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Step 1: Recall that in the frequency interpretation, the probability of an event is defined as the long-term relative frequency of that event occurring in a large number of trials. Show more…

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Prove that $p(A)+p(\neg \mathrm{A})=1$ in the frequency interpretation (without using Kolmogorov's axioms).

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Key Concepts

Frequency Interpretation of Probability

This concept defines the probability of an event as the limit of its relative frequency as the number of trials tends to infinity. That is, if an event A occurs n_A times in n trials, then its probability is p(A) = lim??? (n_A / n). This approach bases probability on observable frequencies rather than abstract axioms.

Complementary Events

An event and its complement are mutually exclusive and collectively exhaustive, meaning that every trial results in either the event A or its complement ¬A. In the frequency interpretation, the relative frequencies of A and ¬A add up to the total number of trials, so when taking the limit as the number of trials grows, p(A) + p(¬A) naturally equals 1.

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