Counting & Probability: Liberal Arts Math Study Notes

Counting & Probability

Introduction to Counting Principles

Counting principles are foundational tools in probability and combinatorics, allowing us to determine the number of ways events can occur. These principles are essential for solving problems involving arrangements, selections, and probability calculations.

• Fundamental Counting Principle: If one event can occur in m ways and a second event can occur in n ways, then the two events together can occur in m × n ways.

• Permutations: The number of ways to arrange n distinct objects in order is given by n! (n factorial).

• Combinations: The number of ways to choose r objects from n without regard to order is given by the binomial coefficient: $\displaystyle \binom{n}{r} = \frac{n!}{r!(n-r)!}$

• Example: The number of ways to select 3 students from a group of 10 is $\displaystyle \binom{10}{3} = 120$.

Probability Basics

Probability quantifies the likelihood of an event occurring, expressed as a number between 0 and 1. The basic formula for probability is:

• Probability of an Event: $\displaystyle P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

• Complementary Events: The probability that event E does not occur is $\displaystyle P(E') = 1 - P(E)$

• Example: If there are 5 red balls and 15 blue balls in a bag, the probability of drawing a red ball is $\displaystyle \frac{5}{20} = 0.25$.

Applications: Probability with Tables

Many probability problems use tables to organize data about possible outcomes. These tables help in identifying the number of favorable and total outcomes for various events.

• Two-way Tables: Used to display the frequency of combinations of two categorical variables.

• Reading Tables: To find the probability of a specific event, sum the relevant frequencies and divide by the total.

• Example: If a table shows the number of students by major and year, the probability of selecting a sophomore math major is the count in that cell divided by the total number of students.

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It is calculated using the formula:

• Conditional Probability: $\displaystyle P(A|B) = \frac{P(A \cap B)}{P(B)}$

• Example: If 8 out of 20 students are female and 3 of those are math majors, the probability that a randomly selected female is a math major is $\displaystyle \frac{3}{8}$.

Permutations and Combinations in Practice

Permutations and combinations are used to solve problems involving arrangements and selections. Permutations are used when order matters, while combinations are used when order does not matter.

• Permutation Formula: $\displaystyle P(n, r) = \frac{n!}{(n-r)!}$

• Combination Formula: $\displaystyle C(n, r) = \frac{n!}{r!(n-r)!}$

• Example: The number of ways to arrange 4 books out of 10 on a shelf is $\displaystyle P(10, 4) = 5040$.

Summary Table: Counting and Probability Formulas

The following table summarizes key formulas used in counting and probability:

Concept	Formula	Description
Permutation	$P(n, r) = \frac{n!}{(n-r)!}$	Arrangements where order matters
Combination	$C(n, r) = \frac{n!}{r!(n-r)!}$	Selections where order does not matter
Probability	$P(E) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$	Basic probability of an event
Conditional Probability	$P(A|B) = \frac{P(A \cap B)}{P(B)}$	Probability of A given B