Probability, Counting Principles, and Descriptive Statistics: Study Guide

Chapter 11: Counting Principles and Probability

Section 1: Fundamental Principle of Counting

The Fundamental Principle of Counting is a basic rule used to determine the total number of possible outcomes for a sequence of independent events. It is foundational for probability and combinatorics.

• Definition: If one event has m possible outcomes and a second independent event has n possible outcomes, the total number of ways both events can occur is m × n.

• Extension: For more than two events, multiply the number of outcomes for each event.

• Example: If you have 3 shirts, 2 pairs of pants, and 2 pairs of shoes, the total number of outfit combinations is $3 \times 2 \times 2 = 12$.

• Example: A restaurant offers 10 appetizers and 15 main courses. The number of two-course meals is $10 \times 15 = 150$.

• Example: For a 10-question multiple choice test with 4 answers each, the number of ways to answer is $4^{10} = 1,048,576$.

• Example: An electronic gate with a 5-digit code (0 cannot be first): $9 \times 10 \times 10 \times 10 \times 10 = 90,000$ possible codes.

Section 2: Permutations

Permutations are arrangements of items where order matters and no item is used more than once. They are used to count the number of possible ordered arrangements.

• Definition: An ordered arrangement of items; order is important.

• Factorial Notation: $n!$ is the product of all positive integers up to n. For example, $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$; $0! = 1$.

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

• Example: 6 jokes to tell: $6! = 720$ ways to order the jokes.

• Example: 20 people, 3 officer positions: $20P3 = \frac{20!}{17!} = 6,480$ ways.

• Permutations with Duplicate Items: $\frac{n!}{p!q!r!}$ where p, q, r are counts of identical items.

• Example: 'ANA': $\frac{3!}{2!} = 3$ distinct permutations.

• Example: 'MISSISSIPPI': $\frac{11!}{4!4!2!} = 34,650$ distinct arrangements.

Section 3: Combinations

Combinations are selections of items where order does not matter. They are used to count the number of possible groups or subsets.

• Definition: An unordered arrangement of items; order is not important.

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

• Example: 3-person committee from 8 people: $8C3 = \frac{8!}{5!3!} = 56$ committees.

• Example: Forming a 5-person committee with 2 Democrats and 3 Republicans: $48C2 \times 50C3 = 1,128 \times 19,600 = 22,108,800$ possible committees.

Section 4: Probability Concepts

Probability quantifies the likelihood of events. It can be theoretical (based on known outcomes) or empirical (based on observed data).

• Theoretical Probability: $P(E) = \frac{\text{number of outcomes in event}}{\text{total number of possible outcomes}}$

• Example: Probability of flipping tails: $P(E) = \frac{1}{2}$

• Example: Probability of rolling a 3 on a die: $P(E) = \frac{1}{6}$

• Empirical Probability: $P(E) = \frac{\text{observed # of times E occurs}}{\text{total # of observed occurrences}}$

Empirical Probability Table

	Married	Not Married	Divorced	Widowed	Total
Male	68	46	11	3	128
Female	69	41	15	11	136
Total	137	87	26	14	264

• Example: Probability of selecting a divorced person: $\frac{26}{264} = 0.10$

• Example: Probability of selecting a female: $\frac{136}{264} = 0.52$

Section 6: Complement, Mutually Exclusive, and Odds

Probability rules help calculate the likelihood of events and their complements, as well as odds.

• Complement Rule: $P(\text{not } E) = 1 - P(E)$

• Example: Probability of not being dealt a queen: $1 - \frac{1}{13} = \frac{12}{13}$

• Mutually Exclusive Events: $P(A \text{ or } B) = P(A) + P(B)$ if A and B cannot occur together.

• Example: Probability of selecting a king or queen: $\frac{4}{52} + \frac{4}{52} = \frac{8}{52} = \frac{2}{13}$

• Not Mutually Exclusive: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

• Example: Probability a baboon enjoys grooming or screeching: $\frac{18}{25} + \frac{16}{25} - \frac{10}{25} = \frac{24}{25}$

• Odds in Favor: $\text{Odds in favor of } E = \frac{P(E)}{P(\text{not } E)}$

• Odds Against: $\text{Odds against } E = \frac{P(\text{not } E)}{P(E)}$

• Example: Odds against winning a raffle with 10 tickets out of 500: $\frac{49}{1}$

• Odds to Probability: If odds are A to B, $P(E) = \frac{A}{A+B}$

• Example: Odds in favor of a horse: 2 to 5, $P(E) = \frac{2}{7}$

Section 7: Independent and Dependent Events, Conditional Probability

Events can be independent (one does not affect the other) or dependent (one affects the other). Conditional probability measures the likelihood of an event given another has occurred.

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

• Example: Probability of 9 girls in a row: $(\frac{1}{2})^9 = \frac{1}{512}$

• Probability of at least one occurrence: $P(\text{at least once}) = 1 - P(\text{never occurs})$

• Dependent Events: $P(A \text{ and } B) = P(A) \times P(B|A)$

• Example: Probability of selecting 2 Spanish-speaking cousins from 10: $\frac{3}{10} \times \frac{2}{9} = \frac{1}{15}$

• Conditional Probability: $P(A|B) = \frac{\text{# of outcomes common to A and B}}{\text{# of outcomes in B}}$

• Example: Probability of selecting a vowel given the letter precedes h: $\frac{2}{7}$

Section 8: Expected Value

Expected value is the average outcome over many trials, calculated by multiplying each possible outcome by its probability and summing the results.

• Formula: $E = \sum (\text{outcome} \times \text{probability})$

• Example: Expected value of a fair die: $E = \frac{1+2+3+4+5+6}{6} = 3.5$

• Example: Expected number of girls in a family with three children: $E = 0 \times \frac{1}{8} + 1 \times \frac{3}{8} + 2 \times \frac{3}{8} + 3 \times \frac{1}{8} = 1.5$

• Example: Game with die: Expected value is $1.50$ per play.

Chapter 12: Descriptive Statistics

Section 1: Introduction to Statistics and Collecting Data

Statistics is the science of collecting, organizing, analyzing, and interpreting data. It is divided into descriptive and inferential statistics.

• Descriptive Statistics: Methods for summarizing and presenting data.

• Inferential Statistics: Methods for making generalizations and drawing conclusions from data.

• Population: The entire group of interest.

• Sample: A subset of the population.

• Representative Sample: A sample that reflects the characteristics of the population.

• Random Sample: Every element has an equal chance of being selected.

Frequency Distribution Table

Age of Maximum Growth	Number of Boys (Frequency)
10	1
11	2
12	5
13	7
14	9
15	6
16	3
17	1
18	1
Total	35

Grouped Frequency Distribution Table

Test Scores	Number of Students (Frequency)
40-49	3
50-59	6
60-69	6
70-79	11
80-89	9
90-99	5

Stem and Leaf Plot

Stem	Leaf
4	7 5 3
5	7 4 6 7 0 9
6	4 3 8 6 7 6
7	5 6 7 9 5 3 6 4 8 1 6
8	2 2 4 8 0 0 1 7 4
9	3 4 2 4 4

Section 2: Measures of Central Tendency

Central tendency measures describe the center of a data set. The most common are mean, median, mode, and midrange.

• Mean: $\text{Mean} = \frac{\sum x}{n}$

• Mean for Frequency Distribution: $\text{Mean} = \frac{\sum x f}{n}$

• Median: The middle value when data is ordered. If n is odd, median is the middle value; if n is even, median is the mean of the two middle values.

• Median Position: $\frac{n+1}{2}$

• Mode: The value that occurs most frequently.

• Midrange: $\text{Midrange} = \frac{\text{lowest value} + \text{highest value}}{2}$

• Example: For grades 52, 69, 75, 86, 86, 92: Mean = 76.67, Median = 80.5, Mode = 86, Midrange = 72.

Section 3: Measures of Variation

Variation measures describe the spread of data. The most common are range and standard deviation.

• Range: $\text{Range} = \text{highest value} - \text{lowest value}$

• Standard Deviation: $s = \sqrt{\frac{\sum (x - \bar{x})^2}{n-1}}$

• Example: For gas prices $1.17, 1.52, 2.79, 2.08$, mean = $1.89$, standard deviation ≈ $0.71$

• Example: For values $4, 2, 11, 7$, mean = $6$, standard deviation ≈ $3.92$

• Interpretation: Smaller standard deviation means less variation; easier to teach a class with smaller deviation.

Section 4: Normal Distribution and Z-Scores

Many data sets follow a normal distribution. The 68-95-99.7 rule describes the spread, and z-scores measure how far a value is from the mean in standard deviation units.

• 68-95-99.7 Rule: 68% of data within 1 SD, 95% within 2 SD, 99.7% within 3 SD.

• Example: Mean height = 70 in, SD = 4 in. 2 SD above mean: $70 + 2 \times 4 = 78$ in.

• Z-score Formula: $z = \frac{\text{data item} - \text{mean}}{\text{standard deviation}}$

• Example: Math test: mean = 60, score = 70, SD = 20, $z = 0.5$; Vocab test: mean = 60, score = 66, SD = 2, $z = 3$

• Percentile: The percentage of scores below a given value.

• Margin of Error: $\frac{1}{\sqrt{n}} \times 100\%$

• Example: For n = 1000, margin of error ≈ 3.2%

Section 5: Applications of Normal Distribution

Normal distribution is used to calculate probabilities and percentages for real-world data.

• Example: Smartphone plan charges: mean = $62, SD = $18$, percentage less than $83.60$ is 88.49% ($z = 1.2$)

• Example: Human pregnancies: mean = 266 days, SD = 16 days, percentage more than 274 days is 30.85% ($z = 0.5$)

• Example: Self-employed work hours: mean = 44.6, SD = 14.4, percentage between 37.4 and 80.6 hours is 68.35% ($z = -0.5$ to $z = 2.5$)