📏Honors Pre-Calculus Unit 11 – Sequences, Probability & Counting Theory

Key Concepts

• Sequences defined as ordered lists of numbers or terms that follow a specific pattern or rule
• Arithmetic sequences have a constant difference between consecutive terms
• Geometric sequences have a constant ratio between consecutive terms
• Probability quantifies the likelihood of an event occurring expressed as a value between 0 and 1
• Sample space represents all possible outcomes of an experiment or event
• Permutations count the number of ways to arrange objects in a specific order
• Combinations count the number of ways to select objects from a group without regard to order
• Expected value measures the average outcome of a random variable over many trials

Sequence Basics

• Sequences can be finite (terminating after a certain number of terms) or infinite (continuing indefinitely)
• The first term of a sequence is denoted as $a_1$, the second term as $a_2$, and so on
• The $n$th term of a sequence, denoted as $a_n$, represents the general formula for any term in the sequence
• To find the next term in a sequence, identify the pattern or rule and apply it to the previous term(s)
• Sequences can be represented visually using graphs, tables, or diagrams
  • Graphs can show the relationship between the term number and its corresponding value
  • Tables organize the terms of a sequence in a structured format
• Recursive formulas define each term of a sequence based on the previous term(s)
• Explicit formulas express the $n$th term of a sequence as a function of $n$

Types of Sequences

• Arithmetic sequences have a constant difference ($d$) between consecutive terms
  • The general formula for the $n$th term of an arithmetic sequence is $a_n = a_1 + (n - 1)d$
  • Example: 2, 5, 8, 11, 14, ... (constant difference of 3)
• Geometric sequences have a constant ratio ($r$) between consecutive terms
  • The general formula for the $n$th term of a geometric sequence is $a_n = a_1 \cdot r^{n-1}$
  • Example: 3, 6, 12, 24, 48, ... (constant ratio of 2)
• Harmonic sequences have terms that are the reciprocals of an arithmetic sequence
  • The general formula for the $n$th term of a harmonic sequence is $a_n = \frac{1}{a + (n - 1)d}$
• Fibonacci sequence is a special sequence where each term is the sum of the two preceding terms
  • The sequence begins with 0 and 1, and the formula is $F_n = F_{n-1} + F_{n-2}$ for $n \geq 2$
  • Example: 0, 1, 1, 2, 3, 5, 8, 13, ...
• Quadratic sequences have second differences (differences between consecutive differences) that are constant
  • The general formula for the $n$th term of a quadratic sequence is $a_n = an^2 + bn + c$

Probability Fundamentals

• Probability is a measure of the likelihood that an event will occur
  • Expressed as a value between 0 (impossible) and 1 (certain)
  • Can also be expressed as a percentage or fraction
• Sample space ($S$) is the set of all possible outcomes of an experiment or event
• An event ($E$) is a subset of the sample space containing one or more outcomes
• The probability of an event $E$ is denoted as $P(E)$ and calculated as $P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}}$
• Complementary events are mutually exclusive and their probabilities sum to 1
  • The complement of event $E$ is denoted as $E'$ or $\overline{E}$
  • $P(E) + P(E') = 1$
• Independent events do not influence each other's outcomes
  • The probability of two independent events $A$ and $B$ occurring is $P(A \cap B) = P(A) \cdot P(B)$
• Dependent events influence each other's outcomes
  • The probability of event $B$ occurring given that event $A$ has occurred is called conditional probability, denoted as $P(B|A)$

Counting Principles

• The Fundamental Counting Principle states that if an event can occur in $m$ ways and another independent event can occur in $n$ ways, then the two events can occur together in $m \times n$ ways
• Permutations count the number of ways to arrange $n$ distinct objects in a specific order
  • The formula for permutations of $n$ objects taken $r$ at a time is $P(n,r) = \frac{n!}{(n-r)!}$
  • Example: The number of ways to arrange 5 books on a shelf is $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$
• Combinations count the number of ways to select $r$ objects from a set of $n$ objects without regard to order
  • The formula for combinations of $n$ objects taken $r$ at a time is $C(n,r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$
  • Example: The number of ways to select 3 students from a group of 10 is $\binom{10}{3} = \frac{10!}{3!(10-3)!} = 120$
• The binomial theorem expands $(a+b)^n$ into a sum of terms involving combinations
  • The general formula is $(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$

Advanced Probability Topics

• Random variables are functions that assign numerical values to outcomes in a sample space
  • Discrete random variables have countable outcomes (e.g., number of heads in 5 coin flips)
  • Continuous random variables have uncountable outcomes (e.g., time until a light bulb burns out)
• Probability distributions describe the likelihood of each possible outcome for a random variable
  • Discrete probability distributions (e.g., binomial, Poisson) assign probabilities to discrete outcomes
  • Continuous probability distributions (e.g., normal, exponential) describe probabilities over a range of values
• Expected value ($E(X)$) is the average value of a random variable $X$ over many trials
  • For a discrete random variable, $E(X) = \sum x \cdot P(X=x)$
  • For a continuous random variable, $E(X) = \int x \cdot f(x) dx$
• Variance ($Var(X)$) measures the spread of a random variable $X$ around its expected value
  • $Var(X) = E((X - E(X))^2) = E(X^2) - (E(X))^2$
• Standard deviation ($\sigma$) is the square root of the variance and measures the typical distance from the mean
  • $\sigma = \sqrt{Var(X)}$

Applications and Problem Solving

• Sequences can model various real-world situations, such as population growth, financial investments, or physical phenomena
  • Example: Compound interest can be modeled using geometric sequences
• Probability is used in fields like genetics, insurance, weather forecasting, and quality control
  • Example: Calculating the probability of inheriting a genetic trait
• Counting principles are applied in areas like cryptography, logistics, and resource allocation
  • Example: Determining the number of possible PIN codes for a 4-digit lock
• Solving sequence and probability problems often involves identifying patterns, applying formulas, and interpreting results
  • Break down complex problems into smaller, manageable steps
  • Use given information to determine the appropriate formula or approach
• Visualizing data through graphs, diagrams, or tables can help in understanding and solving problems
  • Example: Creating a probability tree diagram to calculate the likelihood of multiple events

Connections to Calculus

• Sequences and series are fundamental concepts in calculus
  • Infinite series are the sums of terms in an infinite sequence
  • Convergence tests determine whether an infinite series has a finite sum
• Limits of sequences and series are used to define continuity, derivatives, and integrals
  • The limit of a sequence $\{a_n\}$ is the value $L$ such that $a_n$ approaches $L$ as $n$ approaches infinity
  • The sum of an infinite series is defined as the limit of its partial sums
• Probability theory is the foundation for integral calculus
  • The definite integral can be interpreted as the area under a probability density function
  • Expected value and variance are calculated using integration for continuous random variables
• Calculus techniques, such as differentiation and integration, are used to analyze and optimize probability models
  • Example: Finding the maximum or minimum value of a probability density function