Explain the procedure of assigning probability in the discrete world of probability theory.

In the discrete world of probability theory, the procedure of assigning probability is a fundamental process that involves several logical and mathematical steps to ensure that probabilities are defined consistently, objectively (in some cases), or subjectively (in others), depending on the context. Discrete probability theory deals with experiments that have a finite or countably infinite number of possible outcomes. Assigning probabilities in this framework involves defining a sample space, determining events, and applying axioms and rules of probability.

1. Defining the Sample Space (Ω)

The first step in assigning probabilities is to clearly define the sample space. The sample space, denoted by Ω (omega), is the set of all possible outcomes of a random experiment. Since we are dealing with the discrete case, Ω is either finite or countably infinite.

For example:

• Tossing a coin once: Ω = {Heads, Tails}

• Rolling a die: Ω = {1, 2, 3, 4, 5, 6}

• Selecting a card from a standard deck (without considering suit): Ω = {1, 2, ..., 13}

2. Defining Events

An event is a subset of the sample space. Events can be simple (consisting of a single outcome) or compound (consisting of more than one outcome). For example, in the experiment of rolling a die:

• Simple event: Getting a 3 → {3}

• Compound event: Getting an even number → {2, 4, 6}

3. Approaches to Assigning Probabilities

There are several accepted approaches to assigning probabilities to the elements of the sample space:

a) Classical (A Priori) Approach

This approach is used when all outcomes in the sample space are equally likely. Probabilities are assigned based on the principle of symmetry or fairness.

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