Probability define, basic, types and formula

Introduction to Probability

What is Probability? (Definition & Examples)

Ever wondered how to predict the unpredictable? The Uncertainty Unmasker, a branch of mathematics, steps in to quantify the whim (or caprice) of events. Consider 0 as a dead end and 1 as a guaranteed destination. This analytical tool is an essential asset in fields like statistics, finance, science, and engineering. It assists us in untangling the intricate web of uncertain outcomes.

Key Elements in the Unmasking Process:

• Experiment: An action that unveils an observable result. Imagine chucking a die or giving a coin a toss.
• Payoff: The outcome of a single experiment. Like getting a 3 when you chuck a die.
• Scenario: A collection of payoffs that share a probability score. An example: rolling an even number on a die (which includes the payoffs 2, 4, and 6).
• Outcome Spectrum (S): The entirety of all possible payoffs from an experiment. In a single die chuck, the outcome spectrum is {1, 2, 3, 4, 5, 6}.
• Favorability (P(E)): A measure of how likely a scenario is to unfold, calculated by dividing the number of advantageous payoffs by the total number of possible payoffs.

Unmasking with Different Lenses:

• Theoretical Unmasking: Based on reasoning or models, not actual experiments. For instance, the probability of chucking a 3 on a fair six-sided die is 1/6.
• Experiential Unmasking: Based on real-world experiments and observations. For example, if you toss a coin 100 times and it lands on heads 55 times, the experiential probability of landing on heads is 55/100 or 0.55.
• Intuitive Unmasking: Based on personal judgment or experience rather than precise calculations. For example, a meteorologist could predict a 70% likelihood of rain using their expertise and accessible information.

The Formula for Favorability

For a scenario E within a finite outcome spectrum S:

P(E) = Number of advantageous payoffs / Total number of possible payoffs

Unmasking Examples in Action:

• Coin Toss:
  • Experiment: Tossing a fair coin.
  • Outcome Spectrum (S): {Heads, Tails}
  • Scenario (E): Getting Heads.
  • Favorability (P(E)): Since there is 1 advantageous payoff (Heads) and 2 possible payoffs (Heads or Tails),
  P(Heads) = 1 / 2 = 0.5
• Die Chuck:
  • Experiment: Chucking a fair six-sided die.
  • Outcome Spectrum (S): {1, 2, 3, 4, 5, 6}
  • Scenario (E): Rolling a number greater than 4.
  • Favorability (P(E)): The advantageous payoffs are 5 and 6, so there are 2 favorable outcomes and 6 possible outcomes,
  P(Number > 4) = 2 / 6 = 1/3 ≈ 0.333
• Card Draw:
  • Experiment: Drawing a card from a standard deck of 52 cards.
  • Outcome Spectrum (S): {All 52 cards}
  • Scenario (E): Drawing an Ace.
  • Favorability (P(E)): There are 4 advantageous payoffs (the 4 Aces) and 52 possible outcomes,
  P(Ace) = 4 / 52 = 1/13 ≈ 0.077

Armed with the Enigma Decoder, we can map a path through life’s unpredictabilities. From interpreting the mysterious signals of weather predictions to embarking on pioneering scientific explorations, probability stands as a powerful instrument for navigating the unexplored realms of the uncertain.

Core Concepts of Probability

Events & Sample Space

The realm of potentialities (or outcome panorama) for an experiment, symbolized by S, encompasses every conceivable result. Picture it as an extensive compendium encompassing all imaginable destinations, regardless of their likelihood of realization. In essence, the sample space embodies the entirety of achievable outcomes.

Examples of Possibility Pools:

• Rolling a Fair Die: The possibility pool, S, is {1, 2, 3, 4, 5, 6}.
• Flipping a Coin: The possibility pool, S, is {Heads, Tails}.
• Drawing a Card: The possibility pool, S, is {Ace, 2, 3, …, King} (where each element represents a card).

Zooming in: Events and Subsets

An event is a specific collection of outcomes within the possibility pool, S. Think of it as a highlighted area on the map, encompassing one or more potential destinations.

Examples of Events:

• Die Roll: In rolling a die, the event of getting an even number is the highlighted area, E, containing {2, 4, 6} on the possibility pool.
• Coin Flip: The event of getting heads is the highlighted area, E, containing {Heads} on the possibility pool.
• Card Draw: The event of drawing a face card is the highlighted area, E, containing {Jack, Queen, King} on the possibility pool.

Properties of Highlighted Areas (Events):

• Empty Zone: An event with no outcomes, denoted by Ø. Imagine a completely blank area on the map, representing an impossible outcome.
• Guaranteed Zone: An event encompassing all outcomes in the possibility pool, S. This is like highlighting the entire map, representing a certain event.
• Complementary Zone: The complement of an event, E (denoted Ec), is the remaining area on the map that wasn’t highlighted. It includes all outcomes not in E.
• Non-Overlapping Zones: Two events, E1 and E2, are non-overlapping if they cannot be highlighted simultaneously because they share no common ground.
• Independent Zones: Two events, E1 and E2, are independent if highlighting one area (E1) doesn’t affect the possibility of highlighting another area (E2) on the map.

Understanding possibility pools (sample spaces) and events is essential for calculating probabilities and making predictions in probability and statistics. They are the groundwork for analyzing uncertain outcomes and making informed decisions in real-world situations.

Probability Calculations (Formula & Types)

Within the realm of statistics and mathematics, likelihood assumes a pivotal role in scrutinizing uncertain occurrences and guiding judicious decisions. Grasping the intricacies of likelihood computations, encompassing their equations and categories, proves indispensable across diverse domains, spanning from corporate analytics to scientific inquiry. In this discourse, we shall delve into the rudiments of likelihood computations, examine prevalent equations, and elucidate various classifications of likelihoods.

Introduction to Probability Calculations

Probability is the measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility, and 1 indicates certainty. Probability calculations involve determining the chances of different outcomes in a given scenario.

Common Formulas for Probability Calculations

1. Probability of an Event (P):
   • The likelihood of an event A, denoted by P(A), is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.
   • Formula: P(A)=Number of Favorable Outcomes Total Number of Possible Outcomes P(A) = \frac{{\text{{Number of Favorable Outcomes}}}}{{\text{{Total Number of Possible Outcomes}}}}P(A)=Total Number of Possible Outcomes Number of Favorable Outcomes​
2. Complement of an Event (P’):
   • The complement of an event A, denoted by P'(A) or P(A’), represents the probability of the event not occurring.
   • Formula: P′(A)=1−P(A)P'(A) = 1 – P(A)P′(A)=1−P(A)
3. Intersection of Events (P(A ∩ B)):
   • The likelihood of the intersection of two events A and B represents the likelihood that both events occur simultaneously.
   • Formula (for independent events): P(A∩B)=P(A)×P(B)P(A \cap B) = P(A) \times P(B)P(A∩B)=P(A)×P(B)
4. Union of Events (P(A ∪ B)):
   • The probability of the union of two events A and B represents the likelihood that at least one of the events occurs.
   • Formula (for mutually exclusive events): P(A∪B)=P(A)+P(B)P(A \cup B) = P(A) + P(B)P(A∪B)=P(A)+P(B)

Types of Probability

1. Classical Probability:
   • Based on equally likely outcomes in a sample space.
   • Example: Tossing a fair coin, rolling a fair die.
2. Empirical Probability:
   • Based on observed frequencies from historical data.
   • Example: Determining the probability of rain based on past weather records.
3. Subjective Probability:
   • Based on personal judgment or beliefs.
   • Example: Estimating the likelihood of winning a lottery.

Understanding Different Probabilities

Theoretical vs. Experimental Probability

Likelihood stands as a foundational concept in the realms of mathematics and statistics, furnishing a framework for scrutinizing ambiguity and crafting prognostications. The landscape of likelihood delineates into two principal avenues—Theoretical and Experimental—each offering distinct vantage points and methodologies for gauging the likelihood of events. Within this discourse, we shall unravel the definitions, disparities, and utilities of Theoretical and Experimental Likelihood.

Theoretical Probability

Definition: Theoretical Likelihood, alias Classical Likelihood, finds its moorings in mathematical precepts and theoretical postulations, hinging upon the presumption of equiprobable outcomes within a sample space.

Calculation: The Theoretical Probability of an event A, symbolized by P(A), emerges through the ratio of favorable outcomes to the total feasible outcomes.

Formula: P(A) = \frac{{\text{{Number of Favorable Outcomes}}}}{{\text{{Total Number of Possible Outcomes}}}}

Example: In the case of flipping a fair coin, where heads and tails are equally probable, the theoretical probability of obtaining either outcome is 1/2.

Experimental Probability

Definition: Experimental Probability, or Empirical Probability, finds grounding in observed frequencies derived from practical experiments or real-world datasets, entailing the execution of trials or the collection of empirical data to approximate event likelihoods.

Calculation: The Experimental Probability of an event A, denoted as P(A), is computed as the ratio of event occurrences to total trials or observations.

Formula: P(A) = \frac{{\text{{Number of Times Event A Occurs}}}}{{\text{{Total Number of Trials or Observations}}}}

Example: To ascertain the experimental likelihood of rolling a six on a standard six-sided die, one would conduct multiple rolls and record the occurrences of a six. Dividing the number of successful outcomes by the total number of attempts yields the experimental likelihood.

Divergences between Theoretical and Experimental Probability

Basis: Theoretical Likelihood finds its genesis in mathematical axioms and the presumption of equiprobable outcomes, whereas Experimental Probability emanates from observed frequencies garnered through real-world experimentation or data.

Calculation Method: Theoretical Likelihood is derived through theoretical formulations and assumptions, while Experimental Probability is gleaned via practical experimentation or data collection.

Certainty: Theoretical Probability furnishes precise probabilities within idealized contexts, whereas Experimental Probability furnishes estimations grounded in real-world observations, subject to variability.

Applications

Theoretical Probability: Employed in theoretical mathematics, likelihood theory, and statistical analysis. Applied within educational settings for disseminating and comprehending likelihood concepts.

Experimental Probability: Utilized in scientific inquiry, quality control processes, and risk evaluation. Applied within pragmatic contexts wherein empirical data is accessible, or experiments can be undertaken.

Axiomatic Probability (Optional)

Axiomatic Likelihood, also referred to as Axiomatic Likelihood Theory, lays the groundwork for understanding likelihood by anchoring itself in a collection of axioms or fundamental tenets. This framework provides a meticulous mathematical foundation for likelihood theory, enabling the methodical development of probabilistic concepts and the creation of intricate mathematical structures. Throughout this discussion, we will delve into the essence of Axiomatic Likelihood, closely examine its foundational axioms, and underscore its pivotal role in advancing the exploration of likelihood.

Definition of Axiomatic Probability

Axiomatic Likelihood is a mathematical framework that defines probability in terms of a set of axioms or fundamental principles. It seeks to formalize the concept by establishing a rigorous mathematical structure that governs probabilistic reasoning and inference.

Key Axioms of Axiomatic Probability

1. Non-Negativity Axiom:
   • Probability values are non-negative, meaning they cannot be less than zero.
   • Formally: For any event A, P(A)≥0P(A) \geq 0P(A)≥0.
2. Normalization Axiom:
   • The probability of the entire sample space is equal to 1.
   • Formally: For the sample space S, P(S)=1P(S) = 1P(S)=1.
3. Additivity Axiom:
   • The probability of the union of mutually exclusive events is equal to the sum of their individual probabilities.
   • Formally: For mutually exclusive events A and B, P(A∪B)=P(A)+P(B)P(A \cup B) = P(A) + P(B)P(A∪B)=P(A)+P(B).

Significance of Axiomatic Probability

1. Rigorous Foundation:
   • Axiomatic Likelihood provides a rigorous mathematical foundation for probability theory, ensuring consistency and coherence in probabilistic reasoning.
2. Generalization:
   • By defining likelihood in terms of axioms, Axiomatic Probability allows for the generalization of probabilistic concepts to abstract spaces and complex scenarios.
3. Theoretical Framework:
   • Axiomatic Probability serves as the theoretical basis for advanced probabilistic models and statistical inference techniques used in various fields, including mathematics, statistics, and computer science.
4. Applications:
   • Axiomatic Likelihood is applied in diverse areas, such as physics, finance, engineering, and machine learning, where probabilistic reasoning and uncertainty quantification are essential.

Advanced Probability Techniques

Probability Distributions (Discrete & Continuous) (Optional)

As you deepen your grasp of likelihood, advanced methodologies and concepts become crucial for more intricate analyses and applications. In this section, we will explore probability distributions (both discrete and continuous), conditional likelihood, and Bayes’ Theorem.

Probability Distributions (Discrete & Continuous)

Probability Distributions:
A probability distribution outlines how probabilities are allocated across the potential values of a random variable. There are two primary types: discrete and continuous.

Discrete Probability Distributions:
Discrete distributions apply to cases where the random variable can assume a countable number of distinct values.

Binomial Distribution:
This models the number of successes in a set number of independent Bernoulli trials (each trial has two possible outcomes).
Example: Flipping a coin 10 times and counting the number of heads.
Formula: P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k} p^k (1-p)^{n-k}P(X=k)=(kn​)pk(1−p)n−k, where nnn is the number of trials, kkk is the number of successes, and ppp is the probability of success in each trial.

Poisson Distribution:
This models the number of events occurring in a fixed interval of time or space, given that the events occur at a known constant rate and independently of the time since the last event.
Example: The number of emails received in an hour.
Formula: P(X=k)=λke−λk!P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}P(X=k)=k!λke−λ​, where λ\lambdaλ is the average rate of occurrence.

Continuous Probability Distributions:
Continuous distributions apply to cases where the random variable can take on any value within a specified range.

Normal Distribution:
Also known as the Gaussian distribution, it is symmetric and describes many natural phenomena.
Example: Heights of individuals, IQ scores.
Formula: f(x)=1σ2πe−12(x−μσ)2f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2} \left( \frac{x-\mu}{\sigma} \right)^2}f(x)=σ2π​1​e−21​(σx−μ​)2, where μ\muμ is the mean and σ\sigmaσ is the standard deviation.

Exponential Distribution:
This models the time between events in a Poisson process.
Example: Time until the next earthquake occurs.
Formula: f(x)=λe−λxf(x) = \lambda e^{-\lambda x}f(x)=λe−λx, where λ\lambdaλ is the rate parameter.

Conditional Probability & Bayes’ Theorem

Conditional Probability:
Conditional probability is the likelihood of an event occurring given that another event has already occurred. It is denoted as P(A∣B)P(A|B)P(A∣B), the probability of event A occurring given that event B has occurred.
Formula: P(A∣B)=P(A∩B)P(B)P(A|B) = \frac{P(A \cap B)}{P(B)}P(A∣B)=P(B)P(A∩B)​, provided P(B)>0P(B) > 0P(B)>0.
Example: The probability of drawing an ace from a deck of cards given that a face card has already been drawn.

Bayes’ Theorem:
Bayes’ Theorem is a fundamental principle that relates the conditional and marginal probabilities of random events. It provides a method to update the probability of a hypothesis based on new evidence.
Formula: P(A∣B)=P(B∣A)⋅P(A)P(B)P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}P(A∣B)=P(B)P(B∣A)⋅P(A)​.
Explanation: P(A∣B)P(A|B)P(A∣B) is the probability of event A given event B, P(B∣A)P(B|A)P(B∣A) is the probability of event B given event A, P(A)P(A)P(A) is the prior probability of event A, and P(B)P(B)P(B) is the total probability of event B.
Example: Calculating the probability that a patient has a disease given a positive test result, using the sensitivity and specificity of the test.

Applications:

Medical Diagnosis: Utilizing Bayes’ Theorem to determine the likelihood of a disease given test results.

Spam Filtering: Estimating the probability that an email is spam based on the occurrence of certain words.

Weather Forecasting: Revising weather predictions based on new meteorological data.

Probability Calculations in Action

Probability Calculations in Action

Grasping theoretical concepts is essential, but the true power of likelihood is demonstrated when applied to real-world situations. This section delves into detailed explanations and examples of likelihood calculations, including solved problems and illuminating probability tree diagrams.

Showcasing Probability in Action: Solved Examples

Example 1: Decoding Coin Flips

The Challenge:
What’s the probability of getting exactly two heads in three tosses of a fair coin?

Unveiling the Solution:
Each coin toss is an independent event with two possible outcomes: heads (H) or tails (T). The total number of possibilities when tossing the coin three times is 23=82^3 = 823=8. The favorable outcomes for getting exactly two heads are: HHT, HTH, THH. There are three favorable outcomes.
P(exactly 2 heads)=Number of favorable outcomesTotal number of possible outcomes=38P(\text{exactly 2 heads}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{3}{8}P(exactly 2 heads)=Total number of possible outcomesNumber of favorable outcomes​=83​

Example 2: Demystifying Dice Rolls

The Challenge:
What’s the probability of rolling a sum of 7 with two six-sided dice?

Unveiling the Solution:
Each die has 6 faces, so there are 6×6=366 \times 6 = 366×6=36 possible outcomes when rolling two dice. The favorable outcomes for a sum of 7 are: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). There are six favorable outcomes.
P(sum of 7)=Number of favorable outcomesTotal number of possible outcomes=636=16P(\text{sum of 7}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{6}{36} = \frac{1}{6}P(sum of 7)=Total number of possible outcomesNumber of favorable outcomes​=366​=61​

Example 3: Exploring Card Draws

The Challenge:
What’s the probability of drawing an Ace or a King from a standard deck of 52 cards?

Unveiling the Solution:
There are four Aces and four Kings in a deck, making a total of eight favorable outcomes. The total number of possible outcomes is 52.
P(Ace or King)=Number of favorable outcomesTotal number of possible outcomes=852=213P(\text{Ace or King}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{8}{52} = \frac{2}{13}P(Ace or King)=Total number of possible outcomesNumber of favorable outcomes​=528​=132​

Probability Tree Diagrams: A Visual Aid

Probability tree diagrams act as a visual map, depicting all potential outcomes of an event and facilitating the systematic calculation of probabilities.

Example: Two-Stage Experiment (Coin Toss and Die Roll)

The Challenge:
A coin is tossed, and if it lands on heads, a die is rolled. What’s the probability of getting heads followed by rolling a 3?

Unveiling the Solution:

1. Draw the first level of branches representing the coin toss: Heads (H) and Tails (T).
2. Draw the second level of branches from Heads, representing the possible outcomes of rolling a die: 1, 2, 3, 4, 5, 6.

To calculate the probability of getting heads followed by rolling a 3, multiply the probability of each stage:
P(H and 3)=P(H)×P(3 given H)=12×16=112P(\text{H and 3}) = P(\text{H}) \times P(\text{3 given H}) = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12}P(H and 3)=P(H)×P(3 given H)=21​×61​=121​

These examples illustrate how likelihood theory can be harnessed to solve practical problems, empowering us to make better predictions and understand the spectrum of possible outcomes in various scenarios.

Exploring Probability Further

Delving deeper into the realm of probability offers opportunities to expand your knowledge and expertise in this fascinating field. This section provides resources for further exploration, including video lectures and additional learning materials.

Video Lectures

1. Khan Academy: Probability and Statistics:
   • Khan Academy offers a comprehensive series of video lectures on probability and statistics, covering topics ranging from basic concepts to advanced techniques.
   • The lectures are presented in an accessible and engaging format, making complex ideas easy to understand.
2. MIT OpenCourseWare: Introduction to Probability and Statistics:
   • MIT OpenCourseWare provides free access to lecture videos from MIT’s Introduction to Probability and Statistics course.
   • The course covers fundamental concepts of likelihood theory and their applications in statistics, with lectures delivered by expert professors.
3. Coursera: Probability and Statistics Courses:
   • Coursera offers a variety of online courses on probability and statistics taught by leading instructors from universities around the world.
   • These courses cater to learners of all levels, from beginners to advanced practitioners, and provide interactive learning experiences.

Resources for Learning

1. Textbooks:
   • Explore textbooks on probability theory and statistics written by renowned authors in the field.
   • Recommended titles include “Probability and Statistics for Engineering and the Sciences” by Jay L. Devore and “Introduction to Probability” by Joseph K. Blitzstein and Jessica Hwang.
2. Online Tutorials and Guides:
   • Access online tutorials, guides, and practice problems to reinforce your understanding of probability concepts.
   • Websites like StatTrek, Wolfram Alpha, and Brilliant offer interactive resources for learning probability and statistics.
3. Peer-Reviewed Journals and Research Papers:
   • Stay updated on the latest developments in probability theory by reading peer-reviewed journals and research papers.
   • Journals such as “Probability Theory and Related Fields” and “Journal of Applied Probability” publish cutting-edge research in probability and its applications.

Community and Discussion Forums

1. Online Forums:
   • Join online forums and discussion groups dedicated to probability and statistics.
   • Platforms like Stack Exchange (Mathematics Stack Exchange, Cross Validated) and Reddit (r/probability, r/statistics) provide opportunities to ask questions, share insights, and engage with fellow enthusiasts.
2. Professional Associations:
   • Consider joining professional associations and societies related to probability and statistics, such as the American Statistical Association (ASA) or the Institute of Mathematical Statistics (IMS).
   • These organizations offer networking opportunities, conferences, and resources for professional development.

Statistics & Probability Connection (Optional)

Understanding the relationship between statistics and probability is essential for gaining a comprehensive grasp of both fields. This section explores the connection between statistics and probability, highlighting how they complement each other and are used in tandem to analyze data and make inferences.

Basic Principles of Statistical Analysis

1. Descriptive Statistics:
   • Descriptive statistics involve summarizing and presenting data in a meaningful way, using measures such as mean, median, mode, variance, and standard deviation.
   • Descriptive statistics provide insights into the characteristics and distribution of data, helping to understand its central tendencies and variability.
2. Inferential Statistics:
   • Inferential statistics involve making predictions or inferences about a population based on a sample of data.
   • Techniques such as hypothesis testing, confidence intervals, and regression analysis are used to draw conclusions about population parameters from sample statistics.
3. Probability Distributions in Statistics:
   • Probability distributions play a central role in statistical analysis, as they provide mathematical models for describing the distribution of data.
   • Common probability distributions used in statistics include the normal distribution, binomial distribution, Poisson distribution, and exponential distribution.

Probability in Statistical Inference

1. Sampling Theory:
   • Probability theory is foundational to sampling theory, which governs the process of selecting samples from populations for statistical analysis.
   • Probability distributions, such as the sampling distribution of a statistic, are used to quantify the variability of sample statistics and make inferences about population parameters.
2. Confidence Intervals:
   • Confidence intervals provide a range of values within which a population parameter is likely to lie, based on sample data and probability theory.
   • Likelihood distributions, particularly the normal distribution, are used to calculate confidence intervals for population parameters such as the mean and proportion.
3. Hypothesis Testing:
   • Hypothesis testing involves making decisions about population parameters based on sample data and probability calculations.
   • Likelihood distributions, such as the t-distribution and chi-square distribution, are used to determine the likelihood of observing sample statistics under different hypotheses.

Bayesian Statistics

1. Bayesian Inference:
   • Bayesian statistics is a framework for statistical inference that incorporates prior knowledge or beliefs about parameters into the analysis.
   • Probability theory is central to Bayesian inference, as it provides the mathematical basis for updating prior beliefs in light of new data using Bayes’ Theorem.
2. Bayesian Decision Theory:
   • Bayesian decision theory is a decision-making framework that uses probability theory to optimize decisions under uncertainty.
   • It combines probability distributions, utility functions, and decision criteria to identify the best course of action given available information and preferences.

conclusion

Probability serves as a foundational instrument for comprehending unpredictability and rendering informed choices. Spanning from elementary principles to sophisticated methodologies, it assumes a pivotal position across diverse domains such as science, economics, and daily existence. Proficiency in likelihood empowers individuals to gauge uncertainty, scrutinize data, and formulate forecasts. Continue delving into and employing your expertise to confront novel obstacles.