3.1 Terminology
2 min read•june 25, 2024
is all about predicting outcomes. We use terms like experiments, outcomes, and sample spaces to describe chances of things happening. Understanding these basics helps us calculate probabilities for different scenarios.
Probability calculations involve simple math. We look at how many ways something can happen compared to all possible outcomes. This helps us figure out the likelihood of events occurring alone or together.
Probability Terminology and Concepts
Key probability terminology
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- : process or procedure that generates well-defined outcomes (rolling a die, flipping a coin, drawing a card from a deck)
- : single result of an (rolling a 3 on a die, flipping tails on a coin, drawing the Ace of Spades from a deck)
- : set of all possible outcomes of an experiment, denoted by ( for rolling a die, for flipping a coin)
- : subset of the sample space, collection of one or more outcomes (rolling an even number on a die, flipping heads on a coin, drawing a red card from a deck)
- Random variable: a function that assigns a real number to each in the sample space
Probability calculation for equal events
- events have the same probability of occurring, in a experiment all outcomes are equally likely
- Probability of an : number of favorable outcomes divided by the total number of possible outcomes, formula (probability of rolling a 3 on a fair die is )
- Probability of the sample space is always 1, formula
- : the probability of an event not occurring, denoted as
"And" vs "Or" events
- "AND" events () occur simultaneously, denoted by (rolling a 3 AND flipping heads)
- Formula: if events are independent
- "OR" events () occur when at least one of two or more events happen, denoted by (rolling a 3 OR flipping heads)
- Formula:
- cannot occur simultaneously, if A and B are mutually exclusive then
- Formula for mutually exclusive events:
Additional Probability Concepts
- : the probability of an event occurring given that another event has already occurred, denoted as
- : two events are independent if the occurrence of one does not affect the probability of the other
- : as the number of trials in an experiment increases, the experimental probability approaches the theoretical probability
Key Terms to Review (21)
Complement: In probability and statistics, the complement of an event is the set of all outcomes in a sample space that are not part of that event. Understanding the complement is crucial because it allows for the calculation of probabilities using the rule that states the probability of an event plus the probability of its complement equals one.
Complement of event A: The complement of event A, denoted as $A^c$ or $\overline{A}$, consists of all outcomes in the sample space that are not in event A. It is the opposite of event A happening.
Conditional probability: Conditional probability is the likelihood of an event occurring given that another event has already occurred. This concept is crucial in understanding how probabilities can change based on prior information and is linked to various ideas like independence, mutual exclusivity, and joint probabilities.
Equally likely: Equally likely events are those that have the same probability of occurring. In probability theory, this concept is fundamental when considering outcomes in a uniform distribution.
Event: An event is a specific outcome or a set of outcomes of a random experiment. Events are subsets of the sample space.
Event: An event is a specific outcome or occurrence that can be observed or measured in a statistical study. It represents a particular result or state of interest within a given context or experiment.
Experiment: An experiment is a process or study that results in the collection of data. It aims to discover patterns, test hypotheses, or determine cause and effect relationships.
Experiment: An experiment is a scientific investigation where researchers manipulate variables and observe the effects in a controlled setting. It is a fundamental tool used to test hypotheses and gather empirical evidence about the world around us.
Fair: Fair describes a situation where all possible outcomes are equally likely to occur. In probability, it means each event has an equal chance of happening.
Independence: Independence is a fundamental concept in statistics that describes the relationship between events or variables. When events or variables are independent, the occurrence or value of one does not depend on or influence the occurrence or value of the other. This concept is crucial in understanding probability, statistical inference, and the analysis of relationships between different factors.
Intersection: Intersection refers to the common elements or outcomes shared between two or more sets. In probability and statistics, it is crucial for understanding how events relate to each other, particularly when calculating probabilities involving multiple events. Recognizing intersections helps in applying fundamental rules of probability and visualizing relationships through diagrams.
Law of large numbers: The Law of Large Numbers states that as the sample size increases, the sample mean will get closer to the population mean. This principle is fundamental in probability and statistics.
Law of Large Numbers: The law of large numbers is a fundamental concept in probability theory that states that as the number of independent trials or observations increases, the average of the results will converge towards the expected value or mean of the probability distribution. This principle underlies the predictability of large-scale events and the reliability of statistical inferences.
Long-term relative frequency: Long-term relative frequency is the proportion of times an event occurs over a large number of trials. It represents the event's probability based on empirical evidence.
Mutually Exclusive Events: Mutually exclusive events are events that cannot occur simultaneously. If one event happens, the other event(s) cannot happen at the same time. This concept is fundamental in understanding probability and how to calculate the likelihood of various outcomes.
Outcome: An outcome is the result of a single trial of a probability experiment. In statistics, it represents one possible event in a set of all possible events.
Outcome: In the context of statistics, an outcome refers to the result or consequence of a particular event or experiment. It represents the possible values or states that can be observed or measured from a random process or a set of conditions.
Probability: Probability is the measure of the likelihood of an event occurring. It is a fundamental concept in statistics that quantifies the uncertainty associated with random events or outcomes. Probability is central to understanding and analyzing data, making informed decisions, and drawing valid conclusions.
Sample Space: The sample space, denoted by the symbol $S$, refers to the set of all possible outcomes or results of an experiment or observation. It represents the complete collection of all possible events or scenarios that can occur in a given situation.
Unfair: Unfair describes a situation or event that does not follow the principles of fairness, equality, or justice. In statistics, it often refers to biases or inequalities in sampling, data collection, or analysis.
Union: In probability and set theory, a union refers to the combination of two or more sets, including all elements that are in any of the sets involved. This concept helps in understanding how different groups or events can overlap and provides a way to calculate probabilities when dealing with multiple scenarios. The union is denoted by the symbol '∪', and it is crucial for analyzing relationships between events, especially when visualizing them through diagrams.