Probability: Definition of Probability

Probability is a fundamental concept in mathematics and statistics that quantifies the likelihood or chance of an event or outcome occurring. It provides a numerical measure between 0 and 1, where a probability of 0 indicates impossibility (an event will not occur), and a probability of 1 indicates certainty (an event will definitely occur). The concept of probability serves as the foundation for understanding uncertainty, randomness, and the principles of statistical inference.

Key Components:

1. Sample Space: The set of all possible outcomes or events of an experiment or random phenomenon. It represents the complete set of alternatives from which an outcome will be observed.
2. Event: A subset of the sample space representing a specific outcome or a combination of outcomes. Events can be simple (single outcome) or compound (multiple outcomes).
3. Probability Measure: A function that assigns a numerical value (probability) between 0 and 1 to each event, indicating the likelihood of the event occurring relative to the sample space.

Mathematical Definition:

The probability

$�(�)$

P(A) of an event A occurring is defined as the ratio of the number of favorable outcomes to event

A to the total number of possible outcomes in the sample space

S:

$\mathrm{<span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="mord\; mathnormal">P</span><span\; class="mopen">(</span><span\; class="mord\; mathnormal">A</span><span\; class="mclose">)</span></span></span>}=\frac{\text{Number of Favorable Outcomes for Event }\mathrm{<br>}}{\text{Total Number of Possible Outcomes in }\mathrm{<br>}}$

P(A)=Total Number of Possible Outcomes in SNumber of Favorable Outcomes for Event A​

Properties of Probability:

1. 0 ≤ P(A) ≤ 1: The probability of any event $A$ lies between 0 and 1, inclusive.
2. P(S) = 1: The sum of the probabilities of all possible outcomes in the sample space
   
   $\mathrm{<br>}$

   S is equal to 1.

3. Complementary Events: The probability of the complement of an event
   
   $\mathrm{<br>}$

   A (denoted as A) 
   ${\mathrm{<br>}}^{\mathrm{\prime }}$

    

   
   $\mathrm{<br>}$

    

    

   P(A′)=1−P(A).

4. Addition Rule: For any two events
   
   $\mathrm{<br>}$

   A and
   $\mathrm{<br>}$

    

   B, the probability of either event
   $\mathrm{<br>}$

    

   A or event
   $B$

    

   B occurring is
   $$

   P(A∪B)=P(A)+P(B)−P(A∩B), where
   $\mathrm{<span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="mord\; mathnormal">P</span><span\; class="mopen">(</span><span\; class="mord\; mathnormal">A</span><span\; class="mspace"></span><span\; class="mbin">\cap </span><span\; class="mspace"></span></span><span\; class="base"><span\; class="strut"></span><span\; class="mord\; mathnormal">B</span><span\; class="mclose">)</span></span></span>}$

    

   P(A∩B) is the probability of both events
   $\mathrm{<span\; class="math\; math-inline"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"><span\; class="katex"><span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="mord\; mathnormal">A</span></span></span></span>and </span>}$

   
   $\mathrm{<span\; class="math\; math-inline"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"><span\; class="katex"><span\; class="katex-html"\; aria-hidden="true"><span\; class="base"><span\; class="mord\; mathnormal">B\; </span></span></span></span>occurring.</span>}$

Interpretation and Applications:

• Subjective Probability: Based on personal beliefs, judgments, or opinions, often used in decision-making, risk assessment, and forecasting.
• Frequency Interpretation: Based on empirical observations and long-run relative frequencies, commonly used in empirical sciences, experiments, and statistical analyses.
• Conditional Probability: Represents the probability of an event occurring given that another event has already occurred, often used in conditional statements, dependent events, and Bayesian inference.

Probability is a mathematical measure that quantifies the likelihood or chance of events or outcomes occurring within a sample space. Defined as the ratio of favorable outcomes to the total number of possible outcomes, probability provides a fundamental framework for analyzing uncertainty, making predictions, and understanding the principles of statistics and probability theory. By incorporating concepts such as sample spaces, events, and probability measures, probability theory facilitates rigorous analyses, informed decision-making, and systematic exploration of randomness, variability, and uncertainty in diverse fields, including mathematics, science, engineering, finance, economics, and social sciences.