Glossary

3.1 Basic Probability Concepts and Axioms

3 min read•july 23, 2024

fundamentals are the building blocks of understanding in various scenarios. These concepts help us quantify and analyze the likelihood of different outcomes, from simple coin tosses to complex business decisions.

Sample spaces, events, and probability axioms provide a structured framework for calculating probabilities. By mastering these basics, we can tackle more advanced probability problems and make informed choices in uncertain situations.

Probability Fundamentals

Definition and role of probability

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  • Probability numerically measures the likelihood an will occur
    • Assigns a value between 0 and 1 to each possible outcome
      • 0 indicates impossibility (event will never happen)
      • 1 indicates certainty (event will always happen)
    • Quantifies uncertainty by assigning probabilities to different outcomes (weather forecasting, stock market predictions)
  • Enables decision-making under uncertainty
    • Helps identify the most likely outcomes to inform decisions
    • Allows calculation of expected values to compare different choices (investment options, insurance policies)

Sample space and events

  • (SS) contains all possible outcomes of a random experiment
    • Each outcome is a unique element of the sample space
    • Mutually exclusive outcomes cannot occur simultaneously (rolling a 1 and 2 on a die)
    • includes all possible outcomes (all 6 faces of a die)
  • An event (EE) is a subset of the sample space
    • Can be a single outcome (drawing an ace from a deck) or a collection of outcomes (drawing a red card)
    • The probability of an event is the sum of the probabilities of its constituent outcomes
  • Examples:
    • Tossing a coin: S={H,T}S = \{H, T\}
    • Rolling a die: S={1,2,3,4,5,6}S = \{1, 2, 3, 4, 5, 6\}
    • Drawing a card: S={2♡,3♡,…,K♡,A♡,2♢,…}S = \{2\heartsuit, 3\heartsuit, \ldots, K\heartsuit, A\heartsuit, 2\diamondsuit, \ldots\}

Axioms of probability

  • Axiom 1: Non-negativity
    • The probability of any event EE is non-negative: [P(E)](https://www.fiveableKeyTerm:p(e))≥0[P(E)](https://www.fiveableKeyTerm:p(e)) \geq 0
    • Probabilities cannot be negative (a -10% chance of rain is nonsensical)
  • Axiom 2:
    • The probability of the entire sample space SS is 1: [P(S)](https://www.fiveableKeyTerm:p(s))=1[P(S)](https://www.fiveableKeyTerm:p(s)) = 1
    • The sum of probabilities for all outcomes in SS must equal 1 (100%)
  • Axiom 3:
    • For any two AA and BB, the probability of their union is the sum of their individual probabilities: P(A∪B)=[P(A)](https://www.fiveableKeyTerm:p(a))+P(B)P(A \cup B) = [P(A)](https://www.fiveableKeyTerm:p(a)) + P(B)
    • Mutually exclusive events cannot occur together, so their probabilities are added (probability of rolling a 1 or 2 on a die is the sum of rolling a 1 and rolling a 2)
  • These axioms form the foundation for calculating probabilities in various scenarios
    • Ensure probability values are consistent and coherent (no negative probabilities, all probabilities sum to 1)

Classical vs empirical probability

  • Classical approach ()
    • Assumes all outcomes in the sample space are equally likely
    • Probability of an event AA is the number of favorable outcomes divided by the total number of possible outcomes: P(A)=∣A∣∣S∣P(A) = \frac{|A|}{|S|}
      • ∣A∣|A| denotes the number of elements in event AA
      • ∣S∣|S| denotes the number of elements in the sample space SS
    • Example: In a fair coin toss, P(H)=12P(H) = \frac{1}{2} and P(T)=12P(T) = \frac{1}{2}
  • Empirical approach ()
    • Based on observed data or experimental results
    • Probability of an event AA is the relative frequency of its occurrence in a large number of trials: P(A)=n(A)nP(A) = \frac{n(A)}{n}
      • n(A)n(A) is the number of times event AA occurs
      • nn is the total number of trials
    • As the number of trials increases, the converges to the true probability
    • Example: If a die is rolled 1000 times and "4" is observed 150 times, the empirical probability of rolling a "4" is 1501000=0.15\frac{150}{1000} = 0.15

Key Terms to Review (24)

A posteriori probability: A posteriori probability is the probability of an event occurring after taking into account new evidence or information. This concept emphasizes the updating of probabilities based on observed outcomes, reflecting how beliefs change with new data. A posteriori probabilities are crucial for making informed decisions in uncertain situations, allowing for a dynamic assessment of risk and likelihood as more information becomes available.
A priori probability: A priori probability refers to the likelihood of an event occurring based on theoretical reasoning rather than empirical evidence or experimentation. It is often determined by analyzing the possible outcomes of a situation and assigning probabilities based on logical deduction, making it a key concept in the foundational principles of probability.
Additivity: Additivity refers to the principle that the probability of the union of mutually exclusive events is equal to the sum of their individual probabilities. This concept is crucial as it forms the basis for calculating probabilities in various scenarios, ensuring that when events cannot occur simultaneously, their likelihoods can simply be added together to find the overall probability.
Binomial Distribution: A binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, where each trial has two possible outcomes: success or failure. This distribution is characterized by two parameters: the number of trials (n) and the probability of success in each trial (p). Understanding binomial distribution is essential for making predictions about events based on basic probability concepts, applying it to real-world situations, and assessing process capability in various fields.
Classical probability: Classical probability refers to the method of calculating the likelihood of an event occurring based on the assumption that all outcomes in a sample space are equally likely. This concept is fundamental in probability theory and relies on basic principles that define how probabilities are assigned to events, highlighting the connection between outcomes and their probabilities.
Collectively Exhaustive: Collectively exhaustive refers to a set of outcomes or events that encompass all possible scenarios within a particular context, ensuring that at least one of the outcomes must occur. This concept is fundamental in probability because it allows for the complete analysis of sample spaces, making sure no potential outcome is overlooked when calculating probabilities.
Conditional probability: Conditional probability is the likelihood of an event occurring given that another event has already occurred. This concept is crucial for understanding how the occurrence of one event can affect the probability of another, and it lays the groundwork for more complex applications, including Bayesian inference and independence testing.
Empirical Probability: Empirical probability is the likelihood of an event occurring based on observed data or experimental results rather than theoretical predictions. This approach relies on real-world observations to estimate probabilities, making it particularly useful in fields where controlled experiments may be difficult to implement. It serves as a practical method to assess risks and outcomes by taking into account actual occurrences over time.
Event: An event is a specific outcome or a set of outcomes from a random experiment. It can be described in terms of its probability, which reflects the likelihood of the event occurring based on the underlying sample space. Understanding events is crucial as they form the basis for calculating probabilities and analyzing relationships between different events, especially when considering factors like independence and conditionality.
Forecasting sales: Forecasting sales is the process of estimating future sales revenue based on historical data, market analysis, and other influencing factors. This technique is crucial for businesses to make informed decisions about budgeting, inventory management, and strategic planning. Accurate sales forecasts help companies anticipate customer demand, optimize operations, and improve financial performance.
Mutually exclusive events: Mutually exclusive events are events that cannot occur at the same time. When one event happens, it completely prevents the occurrence of the other event. This concept is fundamental to understanding probability, as it connects to how we calculate the likelihood of different outcomes and the implications of independent events.
Non-negativity axiom: The non-negativity axiom states that the probability of any event is always greater than or equal to zero. This principle ensures that probabilities cannot be negative, reflecting the idea that a likelihood of occurrence cannot be less than nothing. This axiom is foundational in understanding probability measures and helps in ensuring that the sum of probabilities for all possible outcomes equals one.
Normal distribution: Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. This characteristic forms a bell-shaped curve, which is significant in various statistical methods and analyses.
Odds: Odds are a way to express the likelihood of an event occurring compared to it not occurring. In probability, odds can be expressed in two forms: odds in favor and odds against, making it an essential concept for assessing risk and decision-making in uncertain situations. Understanding odds helps connect to broader concepts like probability ratios and risk assessment, which are fundamental for effective analysis in various contexts.
P(a ∪ b): The notation p(a ∪ b) represents the probability of the occurrence of either event A or event B or both. This concept is fundamental in understanding how to combine probabilities of different events and is based on the principle of inclusion-exclusion, which ensures that overlapping probabilities are not counted more than once.
P(a): The notation p(a) represents the probability of a specific event 'a' occurring. In probability theory, this term is used to quantify the likelihood of that event happening within a defined sample space. Understanding p(a) is essential as it connects to concepts like outcomes, events, and the foundational rules of probability that govern how probabilities are calculated and interpreted.
P(e): p(e) refers to the probability of an event 'e' occurring, which is a fundamental concept in probability theory. This probability is quantified as a number between 0 and 1, where 0 indicates that the event cannot occur, and 1 indicates certainty that the event will occur. Understanding p(e) is essential for making predictions and informed decisions based on uncertain outcomes.
P(s): p(s) represents the probability of a specific event 's' occurring in a given sample space. It quantifies how likely it is for that event to happen, with values ranging from 0 (impossible) to 1 (certain). Understanding p(s) is crucial for making informed decisions based on statistical reasoning, as it lays the foundation for calculating probabilities and applying them in various contexts.
Probability: Probability is a measure of the likelihood that a specific event will occur, expressed as a number between 0 and 1. It quantifies uncertainty and helps in making informed decisions based on the chances of various outcomes. Understanding probability is essential for analyzing random phenomena and for applying statistical methods to real-world situations.
Risk: Risk refers to the potential of experiencing a loss or negative outcome associated with an uncertain event. In probability and statistics, risk is often quantified to assess the likelihood of various outcomes, enabling better decision-making under uncertainty. Understanding risk helps in evaluating the potential rewards against possible losses when making choices.
Risk assessment: Risk assessment is the systematic process of identifying, evaluating, and prioritizing risks associated with a decision or investment, allowing organizations to minimize potential negative outcomes. By understanding the likelihood and impact of various risks, stakeholders can make informed decisions that balance potential rewards against possible losses.
Sample space: A sample space is the set of all possible outcomes of a random experiment. It serves as the foundation for probability theory, allowing one to analyze and quantify uncertainty. Each outcome in the sample space is mutually exclusive, meaning that they cannot occur simultaneously, and collectively exhaustive, covering all potential results of the experiment.
Uncertainty: Uncertainty refers to the state of having limited knowledge about an event or outcome, making it impossible to predict with complete confidence. In probability, uncertainty is inherent in the randomness of events and influences decision-making, as it necessitates the use of probability to quantify the likelihood of different outcomes and helps assess risks and benefits.
Unit Measure: A unit measure is a standard quantity used to express the size, amount, or value of something in a consistent way. This concept is essential in probability and statistics as it allows for the comparison of different outcomes and events using a common framework. Understanding unit measures helps in calculating probabilities and makes it easier to interpret data, ensuring accurate communication of results.
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