Probability Formula

Probability refers to the likelihood of an event happening. In various real-life situations, we often need to predict the outcomes of different events. These outcomes can be either certain or uncertain. In such cases, we discuss the chances of an event occurring or not. Probability is widely used in numerous fields, including games, business for making forecasts, and the emerging field of artificial intelligence.

What is Probability?

Probability is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes for a given event. In an experiment with 'n' possible outcomes, let 'x' represent the number of favorable outcomes. The probability of an event can be calculated using the following formula:

Probability (Event) = Favorable Outcomes / Total Outcomes = x/n

Probability Formula

The probability formula assesses the likelihood of an event happening. It represents the ratio of favorable outcomes to all possible outcomes. Here is the probability formula:

where,

• The probability of event 'B' is denoted as P(B).
• The count of favorable outcomes for event 'B' is represented by n(B).
• The total number of possible outcomes in a sample space is denoted as n(S).

Different Probability Formulas

• Addition Rule of Probability: When an event is the combination of two other events, A and B, then the probability of either A or B occurring is calculated as:

  P(A or B) = P(A) + P(B) - P(A ∩ B)

• This can be expressed as:

  P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Probability formula with the complementary rule

When one event is the complement of another event, the probability of not A, denoted as P(not A) or P(A'), is calculated as 1 minus the probability of A:

P(not A) = 1 - P(A)

The total probability of events A and its complement A' is always 1: 1 = P(A) + P(A').

For conditional probability, when event A has already occurred and the probability of event B is needed, it is calculated as the probability of both A and B occurring divided by the probability of A:

P(B | A) = P(A ∩ B) / P(A)

According to the multiplication rule of probability, when two events A and B are independent, the probability of both events occurring together (A ∩ B) is equal to the product of their individual probabilities:

P(A ∩ B) = P(A) ⋅ P(B)

Types of Probability

There can be several viewpoints or sorts of probabilities depending on the nature of the outcome or the method used to determine the likelihood of an event occurring. There are four different types of probabilities:

• Traditional Probability
• Probability in Practice
• Personal Probability
• Probability axiomatic

Traditional Probability

Definition: Traditional probability, also known as classical or theoretical probability, is the likelihood of an event occurring based on all possible outcomes being equally likely. It is often used in games of chance, such as rolling dice or flipping coins.

Formula:

P(A) = (Number of favorable outcomes) / (Total number of possible outcomes)

Example:

When rolling a fair six-sided die, the probability of rolling a 3 is 1/6, since there is one favorable outcome (rolling a 3) out of six possible outcomes (rolling a 1, 2, 3, 4, 5, or 6).

Probability in Practice

Definition: Probability in practice, also known as empirical or experimental probability, is the likelihood of an event occurring based on actual experiments or historical data. It is determined by conducting trials and recording outcomes.

Formula:

P(A) = (Number of times event A occurs) / (Total number of trials)

Example:

If you flip a coin 100 times and it lands on heads 55 times, the empirical probability of getting heads is 55/100 = 0.55.

Personal Probability

Definition: Personal probability, also known as subjective probability, is the individual's degree of belief in the occurrence of an event. It is based on personal judgment, intuition, or opinion rather than on precise calculations or historical data.

Example:

If you believe there is a 70% chance that your favorite sports team will win their next game, that is your personal probability based on your knowledge and intuition about the team’s performance.

Probability Axiomatic

Definition: Axiomatic probability is a formal mathematical approach to probability based on a set of axioms or rules. It provides a rigorous foundation for probability theory, ensuring consistency and logical coherence in probability calculations.

Axioms of Probability:

• Non-negativity: For any event A, the probability of A is greater than or equal to 0. P(A) ≥ 0
• Normalization: The probability of the sample space S is 1. P(S) = 1
• Additivity: For any two mutually exclusive events A and B, the probability of their union is the sum of their individual probabilities. P(A ∪ B) = P(A) + P(B)

Example:

If you have a sample space S containing all possible outcomes of rolling a die, the probability of any specific outcome (such as rolling a 1) is based on the axioms. For example, if A is the event of rolling a 1, then P(A) ≥ 0, P(S) = 1, and the sum of probabilities for all individual outcomes equals 1.