Combinatorics and Probability

Counting Principles and Set Theory

Counting and set theory are foundational topics in probability and combinatorics, allowing us to systematically count outcomes and analyze relationships between sets.

• Union of Sets (A ∪ B): The set of elements in A, B, or both.

• Intersection of Sets (A ∩ B): The set of elements common to both A and B.

• Complement of a Set (Ac): The set of elements not in A.

• Disjoint Sets: Two sets are disjoint if they have no elements in common, i.e., A ∩ B = ∅.

Example: Given the following table:

• n(A ∩ B) = 5

• n(A ∪ B) = 5 + 18 + 9 = 32

• n(A ∩ Bc) = 9

• n(Ac ∩ B) = 18

Venn Diagrams are often used to visually represent set relationships.

Permutations and Combinations

Permutations and combinations are methods for counting arrangements and selections of objects.

• Permutation: An ordered arrangement of objects. The number of permutations of n objects taken r at a time is:

• Combination: An unordered selection of objects. The number of combinations of n objects taken r at a time is:

• Factorial:

Examples:

• Arranging 5 books on a shelf: ways.

• Selecting 3 people from 7: ways.

• Forming a 4-person committee from 12: ways.

• Choosing a president, vice president, and secretary from 10: ways.

Counting with Restrictions

Sometimes, arrangements or selections have restrictions, such as no repetitions or specific groupings.

• Passwords: For a password of 2 digits (0-9) followed by 3 letters (A-Z):

  • If digits and letters can be repeated:

  • If no digits or letters are repeated:

• Committee Selection: If a committee of 6 is to be chosen from two departments (A: 12, B: 9):

  • 3 from A and 3 from B:

  • No employees from B:

Probability Basics

Probability quantifies the likelihood of an event occurring, ranging from 0 (impossible) to 1 (certain).

• Probability of an Event:

• Complement Rule:

• Union of Two Events:

• Conditional Probability:

Example: If two dice are rolled, the probability that the sum is 8:

• Possible outcomes: 36

• Ways to get 8: (2,6), (3,5), (4,4), (5,3), (6,2) = 5 ways

Probability with Cards

Standard deck: 52 cards, 4 suits, 13 ranks per suit.

• Number of 5-card hands with 3 clubs and 2 hearts:

• Number of 5-card hands with all red cards:

• Number of hands with 3 kings and 2 jacks:

• All face cards (12 in deck):

Probability Distributions and Expected Value

Probability distributions assign probabilities to each possible value of a random variable.

• Expected Value (Mean):

Example: Drawing a card from a deck: win $10 if diamond, lose $4 otherwise.

Probability Tables and Contingency Tables

Contingency tables summarize data for two categorical variables.

• Probability a person is a heavy smoker:

• Probability a person is a man and a heavy smoker:

Probability Trees and Conditional Probability

Probability trees help visualize multi-stage experiments, such as drawing balls from a box without replacement.

• Example: Box with 4 red and 2 white balls. Probability of drawing a white ball on the second draw given the first is red:

• First draw red: ; then white: ; so

Binomial and Hypergeometric Distributions

These distributions model the probability of a given number of successes in a fixed number of trials.

• Binomial Distribution: Used when each trial is independent and has the same probability of success.

• Hypergeometric Distribution: Used when sampling without replacement from a finite population.

Example: A box of 10 bulbs, 3 defective. Probability that 2 out of 2 sampled are defective:

Applications in Probability

• Insurance Expected Value:

• Quality Control: Probability that a shipment is rejected if at least one defective is found in a sample.

Summary Table: Key Formulas

Additional info: Some context and explanations have been expanded for clarity and completeness, as the original material was in question format with brief notes and calculations.