Probability Distributions and Expected Value

Overview

This lecture introduces probability distributions, explains how to calculate the expected value, and provides examples using dice, family scenarios, and insurance.

Probability Distributions

• A probability distribution lists all possible outcomes of an event and their associated probabilities.
• For two six-sided dice, the distribution shows the probability of each possible sum (2 through 12).
• The probability of each sum is calculated by dividing the number of ways the sum can occur by the total number of possible outcomes.

Creating Distributions with Random Variables

• A random variable records a numerical outcome from an experiment (e.g., number of girls in a family of three).
• List all possible outcomes for the random variable (0, 1, 2, or 3 girls).
• Calculate the probability for each outcome using the sample space (e.g., P(0 girls) = 1/8).

Expected Value

• Expected value (E[X]) represents the average outcome if an experiment is repeated indefinitely.
• E[X] is calculated by multiplying each outcome by its probability and summing the results.
• Example: Picking a marble with different prize values, E[X] = (probability x value) for each outcome, summed.

Real-world Application: Insurance

• Insurance policies can be analyzed using expected value.
• For an insured bike: outcomes are "loss" ($545 after premium) and "no loss" (-$55 premium), with given probabilities.
• The expected value may be negative but serves as risk management for rare, costly events.

Calculation Formula

• Expected value: E[X] = (X₁ × P₁) + (X₂ × P₂) + (X₃ × P₃) + ... for all possible outcomes.
• Always perform multiplications before additions when calculating.

Key Terms & Definitions

• Distribution — A table or function showing all possible outcomes and their probabilities.
• Random Variable — A variable representing numerical outcomes of a probabilistic event.
• Expected Value (E[X]) — The long-term average result of repeating an experiment.
• Sample Space — The set of all possible outcomes in a probability experiment.

Action Items / Next Steps

• Practice creating probability distributions and calculating expected value for various experiments.
• Review formulas and be prepared to apply them to real-world scenarios, such as insurance.