3 min read•Last Updated on June 27, 2024
Probability events can be independent or mutually exclusive, but rarely both. Independent events don't affect each other's likelihood, while mutually exclusive events can't happen together. Understanding these concepts is crucial for calculating probabilities accurately.
Sampling with and without replacement impacts event independence. With replacement keeps probabilities constant, while without replacement changes them. These ideas form the foundation for more advanced probability concepts like the Law of Total Probability and Bayes' Theorem.
Independence (probability theory) - Wikipedia View original
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Independence (probability theory) - Wikipedia View original
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The addition rule is a fundamental concept in probability theory that allows for the calculation of the probability of the occurrence of one or more mutually exclusive events. It provides a way to determine the probability of the union of two or more events when they are independent or mutually exclusive.
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The addition rule is a fundamental concept in probability theory that allows for the calculation of the probability of the occurrence of one or more mutually exclusive events. It provides a way to determine the probability of the union of two or more events when they are independent or mutually exclusive.
Term 1 of 23
Probability is the measure of the likelihood that an event will occur. It quantifies the chance or possibility of a particular outcome happening within a given set of circumstances or a defined sample space. Probability is a fundamental concept in statistics, as it provides a framework for understanding and analyzing uncertain events and their associated likelihoods.
Sample Space: The set of all possible outcomes or events that can occur in a given situation or experiment.
Event: A specific outcome or set of outcomes that can occur within a sample space.
Mutually Exclusive Events: Events that cannot occur simultaneously or share any common outcomes within a sample space.
Independent events are events whose outcomes do not influence or depend on the outcomes of other events. The occurrence of one event does not affect the probability of the other event occurring.
Mutually Exclusive Events: Mutually exclusive events are events that cannot occur simultaneously. If one event occurs, the other event cannot occur.
Conditional Probability: Conditional probability is the likelihood of an event occurring given that another event has already occurred.
Multiplication Principle: The multiplication principle states that the probability of two independent events occurring together is the product of their individual probabilities.
Mutually exclusive events are events that cannot occur simultaneously or together. If one event happens, the other event(s) cannot happen at the same time. This concept is central to understanding probability and how to calculate the likelihood of events occurring.
Independent Events: Independent events are events where the occurrence of one event does not affect the probability of the other event occurring.
Probability: Probability is the measure of the likelihood that an event will occur, expressed as a number between 0 and 1.
Venn Diagram: A Venn diagram is a visual representation of the relationships between sets or events, often used to illustrate mutually exclusive events.
The law of total probability is a fundamental concept in probability theory that describes how the probability of an event can be calculated by considering the probabilities of mutually exclusive and exhaustive events. It provides a framework for understanding and calculating the probability of an event when the sample space can be divided into distinct, non-overlapping subsets.
Mutually Exclusive Events: Events are mutually exclusive if the occurrence of one event prevents the occurrence of the other event(s). In other words, the events cannot happen simultaneously.
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), which reads as the probability of A given B.
Total Probability: The total probability of an event is the sum of the probabilities of all mutually exclusive and exhaustive events that can lead to that event.
Bayes' Theorem is a fundamental concept in probability and statistics that describes the likelihood of an event occurring given the prior knowledge of the conditions related to that event. It provides a way to update the probability of a hypothesis as more information or evidence becomes available.
Conditional Probability: The probability of an event occurring given that another event has occurred.
Prior Probability: The probability of an event occurring before any new information or evidence is taken into account.
Posterior Probability: The updated probability of an event occurring after new information or evidence is taken into account.
Conditional probability is the likelihood of an event occurring given that another event has already occurred. It represents the probability of one event happening, given the knowledge or occurrence of another related event.
Probability: The measure of the likelihood that an event will occur, expressed as a number between 0 and 1.
Independent Events: Events where the occurrence of one event does not affect the probability of the other event occurring.
Mutually Exclusive Events: Events that cannot occur simultaneously, meaning the occurrence of one event prevents the occurrence of the other event.
Venn diagrams are visual representations that use overlapping circles to illustrate the relationships between different sets or groups. They are commonly used to analyze probabilities, explore logical relationships, and compare and contrast concepts.
Set: A well-defined collection of distinct objects or elements.
Intersection: The set of elements that are common to two or more sets.
Union: The set of all elements that belong to at least one of the sets being considered.
Sampling with replacement is a statistical technique where items are selected from a population, and after each selection, the item is returned to the population before the next selection is made. This means that the same item can be selected multiple times during the sampling process, allowing for the possibility of repetition in the sample.
Sampling without Replacement: In this method, items are selected from a population, and once selected, they are not returned to the population before the next selection is made. This ensures that each item can only be selected once during the sampling process.
Probability Sampling: Probability sampling is a method where each item in the population has a known, non-zero chance of being selected for the sample. This includes techniques like simple random sampling, systematic sampling, and stratified sampling.
Bernoulli Trials: Bernoulli trials are a series of independent experiments, each with two possible outcomes (success or failure), where the probability of success remains constant across all trials.
Sampling without replacement is a method of drawing samples from a population where each item is removed from the population after it is selected, ensuring that no item can be selected more than once. This is in contrast to sampling with replacement, where items are returned to the population after being selected, allowing for the possibility of the same item being chosen multiple times.
Sampling with Replacement: A sampling method where items are returned to the population after being selected, allowing for the possibility of the same item being chosen multiple times.
Hypergeometric Distribution: A discrete probability distribution that describes the probability of a certain number of successes in a fixed number of draws, without replacement, from a finite population.
Independent Events: Events where the occurrence of one event does not affect the probability of the other event occurring.
Hypergeometric probability is a discrete probability distribution that describes the probability of a certain number of successes in a fixed number of trials, without replacement, from a finite population. It is particularly useful in situations where the population size is relatively small, and the probability of success in each trial is not constant across trials.
Probability Distribution: A probability distribution is a mathematical function that describes the possible values and their corresponding probabilities of a random variable.
Discrete Probability Distribution: A discrete probability distribution is a probability distribution where the random variable can only take on a countable number of distinct values.
Finite Population: A finite population is a population that has a known, fixed, and limited number of elements or members.
Independence tests are statistical methods used to determine whether two or more events or variables are independent of each other. They are essential in analyzing the relationships between different phenomena and making informed decisions based on data.
Hypothesis Testing: The process of using statistical analysis to determine whether a particular claim or hypothesis about a population parameter is likely to be true or false.
Chi-Square Test: A statistical test used to determine whether there is a significant difference between the expected and observed frequencies in one or more categories.
Contingency Table: A table that displays the frequency distribution of the variables, allowing for the analysis of the relationship between them.