Probability Rules and Concepts

Basic Probability Rules

Probability is the mathematical study of randomness and uncertainty. The following rules are fundamental for calculating probabilities in various scenarios:

• Total Probability Rule: The probability of the sample space S is always 1.

• Complement Rule: The probability of an event A is 1 minus the probability of its complement.

• Addition Rule for Disjoint Events: If A and B are mutually exclusive,

• General Addition Rule: For any two events,

• Conditional Probability: The probability of B given A is

• General Multiplication Rule:

• Multiplication Rule for Independent Events: If A and B are independent,

Random Variables and Their Properties

Expected Value and Variance

A random variable is a numerical outcome of a random process. The expected value (mean), variance, and standard deviation are key characteristics:

• Expected Value (Mean):

• Variance:

• Standard Deviation:

Combining Random Variables

• Adding/Subtracting Constants: Adding a constant to a random variable shifts the mean but does not affect the variance or standard deviation.

• Multiplying by Constants: Multiplying a random variable by a constant a multiplies the mean by a and the standard deviation by |a|:

• If X and Y are independent:

Binomial Model

Binomial Probability

The binomial model describes the probability of k successes in n independent trials, each with probability p of success:

• Binomial Coefficient:

• Probability of k successes: , where

• Mean (Expected Number of Successes):

• Standard Deviation:

n = number of trials, k = number of successes, p = probability of success, q = probability of failure.

Correlation and Regression

Correlation Coefficient

The correlation coefficient (r) measures the strength and direction of a linear relationship between two quantitative variables:

• Formula:

• Where is the covariance, and are the standard deviations of X and Y, respectively.

Regression Analysis

Regression is used to model the relationship between a dependent variable Y and an independent variable X.

• Residual:

• Line of Best Fit:

• Slope:

• Intercept:

• Regression to the Mean: The predicted value is closer to the mean of Y than X is to the mean of X, unless r = ±1.

• Residual Standard Deviation: Measures the typical size of the residuals (errors) from the regression line.

The Normal Model and Z-Scores

Standard Deviation as a Ruler and the Normal Model

The normal model is a symmetric, bell-shaped distribution characterized by its mean (μ) and standard deviation (σ). Z-scores standardize values for comparison:

• Z-Score:

• Z-scores indicate how many standard deviations a value is from the mean.

• Normal tables (Z-tables) are used to find probabilities and percentiles for standard normal distributions.

Using the Standard Normal Table

The standard normal table provides the area (probability) to the left of a given z-score in the standard normal distribution. To use the table:

• Find the row for the first two digits of the z-score.

• Find the column for the second decimal place.

• The intersection gives the cumulative probability.

• For negative z-scores, use the left table; for positive z-scores, use the right table.

Summary Table: Key Formulas and Concepts