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 solve problems involving arrangements, selections, chance, and expected outcomes in various real-world and mathematical contexts.

Fundamental Counting Principle

Definition and Application

• Fundamental Counting Principle: Used to determine the total number of possible outcomes when there are several choices to make in sequence.

• Multiply the number of 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 cars 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. Use when order is important (e.g., ranking, scheduling).

Formula:

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

Example 1: How many ways can 4 acts be scheduled?

$4! = 4 \times 3 \times 2 \times 1 = 24$

Example 2: 11 applicants for 3 different positions:

$P(11, 3) = \frac{11!}{(11-3)!} = 11 \times 10 \times 9 = 990$

Combinations

Selections Where Order Does Not Matter

• Combination: A selection of items where order does not matter (e.g., choosing a team, selecting books).

Formula:

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

Example: Choosing 4 books from 10:

$C(10, 4) = \frac{10!}{4!6!} = 210$

Permutation vs. Combination Comparison

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

Arrangements with Repeated Letters

Counting Distinct Arrangements

• Used when arranging objects with repeated elements (e.g., letters in a word).

Formula:

$\frac{n!}{a!b!c!}$

where $n$ is the total number of objects, and $a, b, c, \ldots$ are the counts of each repeated object.

Example: Arrangements of "ATLANTA" (7 letters: A=3, T=2, L=1, N=1):

$\frac{7!}{3!2!} = \frac{5040}{12} = 420$

Basic Probability

Calculating Likelihood

• Probability: The likelihood of an event occurring.

Formula:

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

Example: Probability of selecting a freshman from 12 freshmen, 16 sophomores, 20 juniors, and 2 seniors (total 50):

$P(\text{freshman}) = \frac{12}{50} = \frac{6}{25}$

Complementary Probability

Probability of the Opposite Event

• Used when asked for the probability that an event 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:

There are 5 possible ranks (5, 6, 7, 8, 9), each with 4 suits: $5 \times 4 = 20$ cards.

$P = \frac{20}{52} = \frac{5}{13}$

Example: Probability of drawing a black card OR a picture card:

Black cards = 26, Picture cards = 12, Black picture cards = 6 (overlap).

$P = \frac{26 + 12 - 6}{52} = \frac{32}{52} = \frac{8}{13}$

Probability of Multiple Events

Independent and Dependent Events

• For independent events, multiply the probabilities.

• For dependent events (without replacement), adjust the denominator after each selection.

Formula (Independent):

$P(A \text{ and } B) = P(A) \times P(B)$

Example: Probability of spinning red first and blue second on a spinner with 2 red, 2 blue, 2 green, 2 yellow (8 sections):

$P(\text{red first}) = \frac{2}{8}$

$P(\text{blue second}) = \frac{2}{8}$

$P = \frac{2}{8} \times \frac{2}{8} = \frac{1}{16}$

Example (Without Replacement): Drawing two red balls from a box with 5 red, 6 green, 9 yellow (20 total):

First red: $\frac{5}{20}$, Second red: $\frac{4}{19}$

$P = \frac{5}{20} \times \frac{4}{19} = \frac{1}{19}$

Addition Rule: OR

Probability of Either Event Occurring

• Used when a question involves the word "OR".

• Subtract the overlap to avoid double-counting.

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

• Used when the problem states "GIVEN".

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

	Brown	Blue	Green	Total
Male	22	18	10	50
Female	18	20	12	50
Total	40	38	22	100

• Does not have brown eyes: $\frac{60}{100} = \frac{3}{5}$

• Brown or blue eyes: $\frac{40 + 38}{100} = \frac{39}{50}$

• Female or green eyes: $\frac{50 + 22 - 12}{100} = \frac{3}{5}$

• Male, given blue eyes: $\frac{18}{38} = \frac{9}{19}$

• Both people have green eyes (without replacement): $\frac{22}{100} \times \frac{21}{99} = \frac{462}{9900} = \frac{7}{150}$

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 from 20 men, 13 women, 2 others (35 total):

Not a man: 15, Odds against: 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:

$P = \frac{4}{4+1} = \frac{4}{5} = 0.8$

Expected Value

Average Outcome Over Many Trials

• Used for problems involving money, games, prizes, or investments.

Formula:

$E = \sum (x \cdot P(x))$

Example 1: Architect bidding on a theater (cost: $15,000, probability of winning: 0.2, profit if awarded: $80,000):

Net gain if win: $65,000; if lose: -$15,000

$E = (0.2)(65,000) + (0.8)(-15,000) = 13,000 - 12,000 = 1,000$

Meaning: On average, the architect gains $1,000 per bid.

Example 2: Money bag game (cost to play: $20):

Prize	Probability
$1	10/20
$2	5/20
$5	3/20
$10	1/20
$100	1/20

Expected prize:

$E = \frac{10(1) + 5(2) + 3(5) + 1(10) + 1(100)}{20} = \frac{145}{20} = 7.25$

Net expected value: $7.25 - 20 = -12.75$

Meaning: On average, the player loses $12.75 per game.

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 "without replacement," remember the denominator changes after each selection.

• For money problems, subtract the cost to play or participate when calculating net expected value.

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

Additional info: These concepts are foundational for understanding probability and statistics, which are widely applicable in decision-making, risk assessment, and data analysis across disciplines.