- "Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true."
Introduction to the concept of probability, probability distributions, and different types of probability.
Sample space: The set of all possible outcomes of an experiment.
Event: A subset of the sample space.
Probability: The measure of the likelihood of an event occurring.
Adding probabilities: Finding the probability of the union of two events.
Multiplying probabilities: Finding the probability of the intersection of two events.
Conditional probability: The probability of an event given the occurrence of another event.
Independence: Two events are independent if the occurrence of one does not affect the probability of the other.
Bayes' theorem: A formula that allows you to calculate the probability of an event given the occurrence of another event.
Random variables: Variables that take on random values.
Probability distributions: A set of probabilities for each possible outcome of a random variable.
Mean: A measure of the central tendency of a probability distribution.
Variance: A measure of the spread of a probability distribution.
Standard deviation: The square root of the variance.
Bernoulli distribution: A probability distribution that only has two possible outcomes.
Binomial distribution: The probability distribution of the number of successes in a fixed number of independent Bernoulli trials.
Poisson distribution: A probability distribution that models the number of occurrences of an event in a fixed interval of time or space.
Normal distribution: A probability distribution that has a bell-shaped curve.
Central Limit Theorem: A theorem that states that the sampling distribution of the mean of any independent, random variable will be normal or nearly normal, if the sample size is large enough.
Hypothesis testing: The process of using statistics to test a hypothesis about a population.
Confidence intervals: A range of values that is likely to contain the true population parameter.
Probability Space: A probability space is a mathematical model that represents a set of possible outcomes, along with their associated probabilities.
Mutually Exclusive Events: Two events are said to be mutually exclusive if they cannot both occur at the same time. For example, if we toss a coin, the outcomes head and tail are mutually exclusive.
Independent Events: Two events are said to be independent if the occurrence of one event has no effect on the probability of the other event. For example, if we toss a coin twice, the outcome of the second toss is independent of the outcome of the first toss.
Conditional Probability: Conditional probability is the probability of an event occurring given that another event has occurred. For example, the probability of getting a head on the second toss of a coin given that the first toss was a tail.
Bayes' Theorem: Bayes' theorem is a mathematical formula used for calculating conditional probabilities based on prior knowledge of related events.
Random Variables: A random variable is a variable whose value is determined by the outcome of a random event. For example, the number of heads obtained in a series of coin tosses is a random variable.
Probability Distributions: Probability distributions are mathematical models that describe the likelihood of a random variable taking different values, along with their associated probabilities.
Expected Value: Expected value is the sum of the products of each possible outcome and its associated probability. It represents the long-term average value of a random variable.
Variance: Variance is a measure of the spread of a distribution. It represents the average of the squares of the deviation of each outcome from the expected value.
Standard Deviation: Standard deviation is a measure of the amount of variation or dispersion of a distribution. It is the square root of the variance.
- "The probability of an event is a number between 0 and 1."
- "The higher the probability of an event, the more likely it is that the event will occur."
- "A simple example is the tossing of a fair (unbiased) coin... the probability of either 'heads' or 'tails' is 1/2 (which could also be written as 0.5 or 50%)."
- "Probability theory is used widely in areas of study such as statistics, mathematics, science, finance, gambling, artificial intelligence, machine learning, computer science, game theory, and philosophy."
- "These concepts have been given an axiomatic mathematical formalization in probability theory."
- "Probability theory is used... to draw inferences about the expected frequency of events."
- "Probability theory is also used to describe the underlying mechanics and regularities of complex systems."
- "0 indicates impossibility of the event and 1 indicates certainty."
- "Since the coin is fair, the two outcomes ('heads' and 'tails') are both equally probable."
- "The probability of 'heads' equals the probability of 'tails'."
- "...the probability of either 'heads' or 'tails' is 1/2 (which could also be written as 0.5 or 50%)."
- "The probability of either 'heads' or 'tails' is 1/2 (which could also be written as 0.5 or 50%)."
- "Probability theory is used widely in... finance."
- "Probability theory is used widely... in machine learning."
- "Probability theory is used widely... in game theory."
- "Probability theory is used widely... in philosophy to, for example, draw inferences about the expected frequency of events."
- "Probability theory is used widely... in artificial intelligence."
- "Probability theory is used to describe the underlying mechanics and regularities of complex systems."
- "0 indicates impossibility of the event."