Probability and Its Rules

Types of Probability

Probability quantifies the likelihood of events occurring. There are several approaches to defining probability:

• Empirical Probability: Based on observed data from experiments or historical records.

• Theoretical Probability: Calculated using known possible outcomes, assuming all outcomes are equally likely.

• Subjective Probability: Based on personal judgment or experience, not strictly on data or theory.

Probability Rules

• Complement Rule: The probability that an event does not occur is .

• Addition Rule: For disjoint (mutually exclusive) events, .

• Multiplication Rule: For independent events, .

Disjoint Events (Mutually Exclusive)

• Events that cannot occur simultaneously. If and are disjoint, .

Legitimate Assignment of Probabilities

• Probabilities must be between 0 and 1.

• The sum of probabilities for all possible outcomes must equal 1.

General Probability Rules

General Addition Rule

• For any two events:

Conditional Probability

• The probability of event given event has occurred:

General Multiplication Rule

• For any two events:

Independence Criteria

• Events and are independent if and .

Tree Diagram

• A graphical tool to visualize sequences of events and their probabilities.

• Useful for calculating joint and conditional probabilities.

Random Variables

Types of Random Variables

• Discrete Random Variable: Takes on countable values (e.g., number of heads in coin tosses).

• Continuous Random Variable: Takes on any value within an interval (e.g., height, weight).

Probability Model

• A mathematical description of the probabilities associated with all possible values of a random variable.

Expected Value

• The mean of a random variable, representing its long-run average value: for discrete variables.

Standard Deviation and Variance

• Variance:

• Standard Deviation:

Transforming Random Variables

• Shifting: Adding a constant to increases the mean by but does not affect the standard deviation.

• Scaling: Multiplying by a constant multiplies both the mean and standard deviation by .

Combining Random Variables

• For independent random variables and :

• Mean:

• Variance:

Probability Models

Geometric Probability Model

• Models the number of trials until the first success in repeated, independent Bernoulli trials.

• Probability:

Binomial Probability Model

• Models the number of successes in a fixed number of independent Bernoulli trials.

• Probability:

• Success/Failure Condition: Both and for normal approximation.

Poisson Probability Model

• Models the number of events occurring in a fixed interval of time or space, given a constant mean rate.

• Probability:

Uniform Probability Model

• All outcomes are equally likely.

• Probability:

Sampling Distributions and Confidence Intervals for Proportions

Sampling Distribution Model

• Describes the distribution of a statistic (e.g., sample mean, sample proportion) over repeated samples.

Sampling Variability (Sampling Error)

• The natural variation in sample statistics from sample to sample.

Confidence Interval

• A range of values, derived from sample data, that is likely to contain the population parameter.

One-Proportion z-Interval

• Used to estimate a population proportion.

• Formula:

Margin of Error

• The maximum expected difference between the true population parameter and a sample estimate.

Confidence Intervals for Means

Central Limit Theorem (CLT)

• States that the sampling distribution of the sample mean approaches a normal distribution as sample size increases, regardless of the population's distribution.

Student's t-Distribution and Degrees of Freedom (DF)

• Used when the population standard deviation is unknown and sample size is small.

• Degrees of freedom:

One-Sample t-Interval for the Mean

• Formula:

Testing Hypotheses

Hypothesis Testing Concepts

• Null Hypothesis (): The default assumption (e.g., no effect, no difference).

• Alternative Hypothesis (): The claim being tested (e.g., there is an effect).

P-Value

• The probability of observing data as extreme as, or more extreme than, the observed data, assuming is true.

Types of Alternatives

• Two-Sided Alternative: Tests for difference in either direction.

• One-Sided Alternative: Tests for difference in a specific direction.

One-Proportion z-Test

• Tests a hypothesis about a population proportion.

• Test statistic:

Effect Size

• Measures the magnitude of a difference, independent of sample size.

One-Sample t-Test for the Mean

• Test statistic:

More About Tests and Intervals

Statistical Significance and Significance Level

• Statistically Significant: When the p-value is less than the chosen significance level (), the result is considered statistically significant.

• Alpha Level (): The threshold for statistical significance, commonly set at 0.05.

• Critical Value: The value that the test statistic must exceed to reject .

Errors in Hypothesis Testing

• Type I Error: Rejecting when it is true (false positive). Probability = .

• Type II Error: Failing to reject when is true (false negative). Probability = .

• Power: Probability of correctly rejecting when is true. .