Probability: Basic Concepts

Introduction to Probability

Probability quantifies how likely an event is to occur, expressed as a value between 0 and 1. The set of all possible outcomes is called the sample space.

• Probability of an event (P):

• Sample Space Example: Flipping a coin yields S = {Heads, Tails}

• Types of Probability:

  • Theoretical Probability: Based on what could happen; calculated before events occur.

  • Empirical Probability: Based on what did happen; calculated after events occur.

• Example: Rolling a six-sided die, probability of rolling a number greater than 3: S = {1, 2, 3, 4, 5, 6}, favorable outcomes = {4, 5, 6}, so

Complementary Events

Complements and Their Probabilities

The complement of an event A (written as A', Ac, or not A) consists of all outcomes where A does not occur. The total probability of all possible events is always 1.

• Complement Rule:

• Example: Probability of not rolling a 4 on a die:

• Application: Probability of not drawing a queen from a deck:

Addition Rule for Probability

Mutually Exclusive Events

Events are mutually exclusive if they cannot occur at the same time. The probability of either event A or B occurring is the sum of their individual probabilities.

• Addition Rule (Mutually Exclusive):

• Example: Probability of rolling a 3 or a 5:

• Practice: Probability of selecting an ace or a king from a deck:

Non-Mutually Exclusive Events

For events that are not mutually exclusive, both can occur together. The addition rule must subtract the probability of their intersection to avoid double-counting.

• Addition Rule (General):

• Example: Probability of rolling a number greater than 3 or an even number on a die.

Multiplication Rule for Probability

Independent Events

Events are independent if the occurrence of one does not affect the other. The probability of both events A and B occurring is the product of their probabilities.

• Multiplication Rule (Independent):

• Example: Probability of getting heads on two consecutive coin flips:

Dependent Events

Events are dependent if the outcome of one affects the probability of the other. The probability of both events is the product of the probability of the first event and the conditional probability of the second event given the first.

• Multiplication Rule (Dependent):

• Example: Drawing and keeping a blue marble, then drawing a red marble from a bag.

Conditional Probability

Definition and Calculation

Conditional probability is the probability of event B occurring given that event A has occurred.

• Formula:

• Example: Probability that a student has a math major given they have a science major.

Bayes' Theorem

Bayes' Theorem for Conditional Probability

Bayes' Theorem allows calculation of conditional probabilities when direct information is unavailable. It is especially useful in medical testing and diagnostic scenarios.

• Bayes' Theorem:

• Example: Probability that a marble came from the Left Bag given it is red.

Fundamental Counting Principle

Counting Outcomes

The Fundamental Counting Principle states that if there are m ways to do one thing and n ways to do another, there are m × n ways to do both.

• Formula: Total outcomes = (number of options for first event) × (number of options for second event) × ...

• Example: 3 shirts and 4 pants yield possible outfits.

Permutations and Combinations

Permutations

Permutations are arrangements of objects where order matters. The formula for the number of permutations of r objects from n options is:

• Formula:

• Example: Arranging 5 shirts for 5 days:

Permutations of Non-Distinct Objects

When objects are not all distinct, divide by the factorials of identical items.

• Formula:

• Example: Arranging the letters in BANANA:

Combinations

Combinations are selections of objects where order does not matter. The formula for the number of combinations of r objects from n options is:

• Formula:

• Example: Choosing 2 flavors from 32:

Permutations vs. Combinations

• Permutations: Order matters; used for arrangements.

• Combinations: Order does not matter; used for groupings.

• Example: Forming a team from a group is a combination; lining up people is a permutation.

Contingency Tables

Finding Probabilities from Contingency Tables

A contingency table displays frequencies for two categorical variables. Probabilities can be calculated as marginal, joint, or conditional.

• Marginal Probability: Probability of an entire row or column.

• Joint Probability: Probability of two events happening together.

• Conditional Probability: Probability of one event given another has occurred.

Example: Probability a randomly selected student is a senior and drives a car:

Summary Table: Probability Rules