Probability Basics

Probability

Basic Concepts

• Experiment: an action or situation that can produce a set of outcomes
• Outcome: a specific result of an experiment
• Sample Space: the set of all possible outcomes of an experiment
• Event: a set of outcomes of an experiment

Probability Measures

• Probability: a number between 0 and 1 that represents the likelihood of an event occurring
• Theoretical Probability: the probability of an event occurring based on the number of favorable outcomes divided by the total number of outcomes
• Experimental Probability: the probability of an event occurring based on repeated trials of an experiment

Rules of Probability

• Complement Rule: P(A') = 1 - P(A)
• Addition Rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
• Multiplication Rule: P(A ∩ B) = P(A) * P(B|A)

Statistics

Descriptive Statistics

• Measures of Central Tendency:
  • Mean: the average value of a dataset
  • Median: the middle value of a dataset when in order
  • Mode: the most frequently occurring value in a dataset
• Measures of Variability:
  • Range: the difference between the largest and smallest values in a dataset
  • Interquartile Range (IQR): the difference between the 75th percentile and the 25th percentile
  • Variance: the average of the squared differences from the mean
  • Standard Deviation: the square root of the variance

Inferential Statistics

• Hypothesis Testing:
  • Null Hypothesis (H0): a statement of no effect or difference
  • Alternative Hypothesis (H1): a statement of an effect or difference
  • Test Statistic: a value calculated from sample data to determine whether to reject the null hypothesis
  • P-Value: the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true
• Confidence Intervals:
  • Margin of Error: the maximum amount by which the sample statistic may differ from the population parameter
  • Confidence Level: the probability that the interval contains the population parameter

Probability

Basic Concepts

• An experiment is an action or situation that can produce a set of outcomes
• An outcome is a specific result of an experiment
• A sample space is the set of all possible outcomes of an experiment
• An event is a set of outcomes of an experiment

Probability Measures

• Probability is a number between 0 and 1 that represents the likelihood of an event occurring
• Theoretical probability is calculated by dividing the number of favorable outcomes by the total number of outcomes
• Experimental probability is calculated by repeating an experiment multiple times and finding the frequency of the event

Rules of Probability

• The complement rule states that the probability of an event not occurring is 1 minus the probability of the event occurring
• The addition rule states that the probability of two events occurring is the sum of their individual probabilities minus the probability of their intersection
• The multiplication rule states that the probability of two events occurring together is the product of their individual probabilities, with the second event conditional on the first

Statistics

Descriptive Statistics

• Measures of central tendency describe the middle value of a dataset
• The mean is the average value of a dataset
• The median is the middle value of a dataset when in order
• The mode is the most frequently occurring value in a dataset
• Measures of variability describe the spread of a dataset
• The range is the difference between the largest and smallest values in a dataset
• The interquartile range (IQR) is the difference between the 75th percentile and the 25th percentile
• Variance is the average of the squared differences from the mean
• Standard deviation is the square root of the variance

Inferential Statistics

• Hypothesis testing involves testing a null hypothesis against an alternative hypothesis
• The null hypothesis states that there is no effect or difference
• The alternative hypothesis states that there is an effect or difference
• A test statistic is a value calculated from sample data to determine whether to reject the null hypothesis
• The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true
• Confidence intervals estimate a population parameter based on a sample of data
• The margin of error is the maximum amount by which the sample statistic may differ from the population parameter
• The confidence level is the probability that the interval contains the population parameter