Demonstrating Axiomatic Probability Axioms with a Six-Sided Fair Die
Axiomatic probability, grounded in mathematical principles, is the foundation of probability theory. We’ll demonstrate this concept in R using a simple example involving a six-sided fair die. The three axioms of probability—non-negativity, normalization, and additivity—are key to understanding this.
R
# Sample space S for a six-sided fair diesample_space <- 1:6# Probability of each outcomeprob_each_outcome <- rep(1/6, 6)# Non-Negativity Axiom (already satisfied)# Normalization Axiomsum_of_probabilities <- sum(prob_each_outcome)cat("Sum of probabilities in the sample space S:", sum_of_probabilities, "\n")# Additivity Axiom# Let's consider two events: rolling an even number and rolling an odd numberevent_even <- c(2, 4, 6)event_odd <- c(1, 3, 5)# Probability of the union of two eventsprobability_union <- sum(prob_each_outcome[event_even]) + sum(prob_each_outcome[event_odd])cat("Probability of rolling an even or an odd number:", probability_union, "\n") |
Output :
Sum of probabilities in the sample space S: 1
Probability of rolling an even or an odd number: 1
In this code, we define a sample space for a fair six-sided die and assign equal probabilities to each outcome, satisfying the non-negativity and normalization axioms. Then, we calculate the probability of rolling an even or an odd number, demonstrating the additivity axiom. The output shows that the sum of probabilities equals 1, as per the normalization axiom, and the probability of the union of the two events equals 1, consistent with the additivity axiom.
The line “Sum of probabilities in the sample space S: 1” demonstrates that the probabilities assigned to each outcome in the sample space (a fair six-sided die) sum up to 1, satisfying the normalization axiom.
The line “Probability of rolling an even or an odd number: 1” shows that when we calculate the probability of rolling either an even or an odd number (two mutually exclusive events), the result is 1, fulfilling the additivity axiom.
Axiomatic Probability in R
Axiomatic probability, also known as the measure-theoretic or Kolmogorov’s probability, is a foundational approach to probability theory that establishes a rigorous mathematical framework for understanding random events and uncertainty. It was developed by the Russian mathematician Andrey Kolmogorov in the 1930s and has become the standard framework for modern probability theory.
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