Counting, Probability, and Expected Value: Study Guide for Liberal Arts Math

Counting, Probability, and Expected Value

Introduction

This study guide covers essential concepts in counting, probability, and expected value, which are foundational topics in Liberal Arts Mathematics. Understanding these principles enables students to analyze situations involving chance, make informed decisions, and solve real-world problems involving uncertainty.

Fundamental Counting Principle

Definition and Application

• Fundamental Counting Principle: Used to determine the total number of possible outcomes when there are multiple choices for each category.

• Formula: $n_1 \times n_2 \times n_3 \cdots$

• Example: If a car comes in 10 colors, 2 transmission types, 2 four-wheel drive options, 2 air conditioning options, and 3 speaker choices, the total number of different car options is $10 \times 2 \times 2 \times 2 \times 3 = 240$.

Permutations

Arrangements Where Order Matters

• Permutation: An arrangement of objects in a specific order. Used when order is important (e.g., ranking, scheduling).

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

• Factorial (!): The product of all positive integers up to a given number. For example, $4! = 4 \times 3 \times 2 \times 1 = 24$.

• Example 1: Scheduling 4 acts: $4! = 24$ ways.

• Example 2: Arranging 7 airplanes with the first fixed: $6! = 720$ ways.

• Example 3: Assigning 3 different jobs to 11 applicants: $P(11, 3) = 11 \times 10 \times 9 = 990$ ways.

Combinations

Selections Where Order Does Not Matter

• Combination: Selection of items where order is not important (e.g., choosing a group).

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

• Example: Choosing 4 books from 10: $C(10, 4) = \frac{10!}{4!6!} = 210$ possible sets.

• Comparison Table:

Permutation	Combination
Arranging people	Selecting people
Scheduling acts	Choosing books
Assigning jobs	Picking a team
Order matters	Order does not matter

Arrangements with Repeated Letters

Counting Distinct Arrangements

• When to Use: Arranging objects with repeated elements (e.g., letters in a word).

• Formula: $\frac{n!}{a!b!c!}$, where $a, b, c$ are counts of each repeated item.

• Example: "ATLANTA" (7 letters: A=3, T=2, L=1, N=1): $\frac{7!}{3!2!} = 420$ distinct arrangements.

Basic Probability

Calculating Likelihood of Events

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

• Example: Probability of selecting a freshman from a group of 50 students (12 freshmen): $P(\text{freshman}) = \frac{12}{50} = \frac{6}{25}$

Complementary Probability

Probability of the Opposite Event

• Complement: The probability that event A does not occur.

• Formula: $P(\text{not A}) = 1 - P(A)$

• Example: Probability of not selecting a sophomore (16 sophomores out of 50): $P(\text{not sophomore}) = 1 - \frac{16}{50} = \frac{17}{25}$

Probability with a Deck of Cards

Standard Deck Facts and Examples

Type	Number
Total cards	52
Hearts	13
Diamonds	13
Clubs	13
Spades	13
Black cards	26
Red cards	26
Picture cards (J, Q, K)	12

• Example: Probability of drawing a card greater than 4 and less than 10: $\frac{20}{52} = \frac{5}{13}$

• Example: Probability of drawing a black card or a picture card: $\frac{26 + 12 - 6}{52} = \frac{32}{52} = \frac{8}{13}$

Probability of Multiple Events

Independent and Dependent Events

• Independent Events: Events where the outcome of one does not affect the other.

• Formula (Independent): $P(A \text{ and } B) = P(A) \times P(B)$

• Example: Probability of spinning red first and blue second on an 8-section spinner (2 red, 2 blue): $\frac{2}{8} \times \frac{2}{8} = \frac{1}{16}$

• Without Replacement: The total number of items decreases after each selection.

• Example: Drawing two red balls from a box (5 red, 6 green, 9 yellow): $\frac{5}{20} \times \frac{4}{19} = \frac{1}{19}$

Addition Rule: OR

Probability of Either Event Occurring

• Formula: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

• Example: Probability of being a freshman or female in a group of 50 students (25 freshmen, 20 females, 15 female freshmen): $P = \frac{25 + 20 - 15}{50} = \frac{30}{50} = \frac{3}{5}$

Conditional Probability

Probability Given a Condition

• Formula: $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$

• Example: Probability a student is male, given they have blue eyes (18 males with blue eyes out of 38 blue-eyed students): $P(\text{male|blue}) = \frac{18}{38} = \frac{9}{19}$

Probability with Tables

Using Tabular Data for Probability

Question	Work	Answer
Does not have brown eyes	60/100	3/5
Brown or blue eyes	(40+38)/100	39/50
Female or green eyes	(50+22-12)/100	3/5
Male, given blue eyes	18/38	9/19

• Example: Probability both selected people have green eyes (22 green-eyed students out of 100): $\frac{22}{100} \times \frac{21}{99} = \frac{462}{9900} = \frac{7}{150}$

Odds

Comparing Favorable and Unfavorable Outcomes

• Odds in Favor: $\frac{\text{Favorable}}{\text{Unfavorable}}$

• Odds Against: $\frac{\text{Unfavorable}}{\text{Favorable}}$

• Example: Odds against selecting a man (20 men, 15 not men): $15:20 = 3:4$

• Example: If odds against a candidate are 1 to 4, odds in favor are 4:1, and probability of winning is $\frac{4}{5} = 0.8$

Expected Value

Average Outcome Over Many Trials

• Formula: $E = \sum (x \cdot P(x))$

• Steps:

  1. Multiply each possible outcome by its probability.

  2. Add all the results.

• Example 1: Architect bidding on a theater (win: $65,000$ with probability $0.2$, lose: $-15,000$ with probability $0.8$): $E = (0.2)(65,000) + (0.8)(-15,000) = 1,000$

• Example 2: Money bag game (cost to play $20$, expected prize $7.25$): Net expected value $= 7.25 - 20 = -12.75$

Formula Cheat Sheet

Topic	Formula
Counting principle	Multiply choices
Permutations	$P(n, r) = \frac{n!}{(n - r)!}$
Combinations	$C(n, r) = \frac{n!}{r!(n - r)!}$
Repeated letters	$\frac{n!}{a!b!}$
Basic probability	$\frac{\text{Favorable}}{\text{Total}}$
Complement	$1 - P(A)$
Independent events	$P(A) \times P(B)$
Addition rule	$P(A) + P(B) - P(A \cap B)$
Conditional probability	$\frac{P(A \cap B)}{P(B)}$
Expected value	$\sum xP(x)$

Test Tips

• Look for keywords: arrange, schedule, order = permutation; choose, select, group = combination; AND = multiply; OR = addition rule; NOT = complement; GIVEN = conditional probability; average winnings/losses = expected value.

• Always simplify fractions unless a decimal or percentage is requested.

• For problems without replacement, remember the total changes after each selection.

• For expected value in money problems, subtract the cost to play or participate.

• Show your work for partial credit, even if calculations are incorrect.

Additional info: This guide expands on the original study guide by providing full academic explanations, formulas in LaTeX, and structured tables for comparison and reference. It is suitable for exam preparation in a Liberal Arts Math course.