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Types of Probability

There are four major types of probabilities:

• Classical or Theoretical Probability
• Empirical or Experimental Probability
• Subjective Probability
• Axiomatic Probability

Theoretical Probability

Theoretical Probability is the probability calculated using mathematical reasoning, not experiments or real data. It is found by dividing the number of favorable outcomes by the total number of possible outcomes, assuming all outcomes are equally likely.

For example, the theoretical probability of getting a head when flipping a fair coin is 1/2

Empirical or Experimental Probability
Experimental probability is found by performing an experiment and observing the results. It is calculated by dividing the number of times an event occurs by the total number of trials.

For example, if a coin is tossed 10 times and heads appears 6 times, the experimental probability of getting heads is .

Subjective Probability

This is fully based on personal, intuitive, or expert judgment rather than data or calculations.

Example; What is the probability it will rain tomorrow? You may look at several forecasts, uses your expert knowledge of the region, and reports the probability that it will rain tomorrow is 80%.

Axiomatic Probability

Kolmogorov proposed a formal approach based on a set of rules (axioms) that define the properties of probability:

Kolmogorov’s Three Axioms of Probability:

The following rules were proposed by Kolmogorov for probability:

1. Non-negativity: The probability of any event is always zero or positive.   P(A)≥0
2. Normalization: The probability of the sample space (all possible outcomes) is equal to 1. P(S)=1 
3. Additivity: If two events cannot occur at the same time (mutually exclusive), then the probability that either event occurs is the sum of their probabilities.

           P(A∪B)=P(A)+P(B)

           P(A∪B)=P(A)+P(B)

Axiomatic probability is the method of calculating probability using Kolmogorov’s axioms, which apply to all situations. In this approach, probabilities are assigned to events in a logical and consistent way to measure the chance of events occurring or not occurring.

Example of Additivity Axiom: Suppose A fair six-sided die are rolled. Let event A = getting an even number {2, 4, 6}. What is the probability of this event?

• According to Kolmogorov’s axioms:
1. Non-negativity: P(A)≥0
2. Normalization: Total outcomes = 6, so. P(S)=1
3. Additivity: Events {2}, {4}, and {6} are mutually exclusive.

Since each outcome has probability 1/6​: 

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