Probability and Statistics: Fundamental Counting Principles, Permutations, and Combinations

Probability and Statistics

Introduction to Probability

• Most people have an intuitive understanding of probability.
  • Example: A 20% chance of rain implies an 80% chance of no rain.
  • Flipping a coin: Probability of heads is generally understood as 1/2.
• Calculating probabilities can become complex, requiring counting of large sets.
  • Examples: Number of possible social security numbers, license plate numbers.
  • Determining probability of a license plate with numbers in increasing order.

Fundamental Principle of Counting

• The core principle for counting large sets efficiently.
• Example: Outfit Combinations
  • 2 pairs of slacks (brown, black).
  • 3 shirts (yellow, white, blue).
  • 2 ties (red, green).
  • Question: How many different outfits can be created?
• Using a TI Nspire calculator to visualize combinations.
  • Slacks: 2 options.
  • Shirts: 3 options.
  • Ties: 2 options.
  • Total outfits: $2 \times 3 \times 2 = 12$
• Each decision multiplies the possibilities.

Explanation

• If there are $m$ ways to do one thing, and $n$ ways to do another, then there are $m \times n$ ways to do both.
• The total number of outcomes is the product of the number of ways each independent choice can be made.

Example Probability

• Probability of wearing brown slacks, a yellow shirt, and a red tie:
  $\frac{1}{12}$
  (Only 1 specific combination out of 12 possible outfits).

Arrangements and Factorials

• Problem: Arranging three books (A, B, C) on a shelf.
  • Listing possible arrangements: ABC, ACB, BAC, BCA, CAB, CBA (6 arrangements).
  • Becomes impractical with larger numbers (e.g., 20 books).
• Applying the Fundamental Principle of Counting.
  • First book: 3 choices.
  • Second book: 2 choices.
  • Third book: 1 choice.
  • Total arrangements: $3 \times 2 \times 1 = 6$
• Generalization: Arranging $n$ items.
  • $n$ choices for the first spot, $n-1$ for the second, and so on.
  • Total arrangements: $n \times (n-1) \times (n-2) \times … \times 1$

Factorial Notation

• Definition: The product of all positive integers less than or equal to $n$.
• Notation: $n!$
• Example:
  $3! = 3 \times 2 \times 1 = 6$
  $20! = 20 \times 19 \times 18 \times … \times 1$ (a very large number).

Permutations

• Problem: Choosing officers (president, vice president, secretary) from a club of 10 members.
  • How many different slates of officers are possible?
• Applying the Fundamental Principle of Counting.
  • President: 10 choices.
  • Vice President: 9 choices (one person is already president).
  • Secretary: 8 choices (two people are already chosen).
  • Total slates: $10 \times 9 \times 8 = 720$

Permutation Notation

• Definition: An ordered subset.
• Notation: $<em>{n}P</em>{k}$, where:
  • $n$ is the total number of items in the set.
  • $k$ is the number of items being chosen.
• Example: Choosing 3 officers from 10 members: $<em>{10}P</em>{3}$
• Order matters: Bob, Alice, Doug is a different slate than Doug, Alice, Bob.

Calculation Rule

• Start with $n$ and decrease by 1 until you have $k$ numbers.
• Example: $<em>{6}P</em>{4} = 6 \times 5 \times 4 \times 3 = 360$

Combinations

• Problem: Choosing a committee from a club.
• Key Difference: Order does not matter.
  • Committee AB is the same as committee BA.
• Definition: An unordered subset.
• Notation: $<em>{n}C</em>{k}$

Example

• Choosing two officers from five people (A, B, C, D, E).
  • Permutations (ordered): AB, BA, AC, CA, AD, DA, AE, EA, BC, CB, BD, DB, BE, EB, CD, DC, CE, EC, DE, ED (20 permutations).
• Choosing a committee of two from five people (A, B, C, D, E).
  • Combinations (unordered): AB, AC, AD, AE, BC, BD, BE, CD, CE, DE (10 combinations).

Choosing Permutation or Combination

• Crucial first step: Determine if order is important.
  • If order matters: Permutation.
  • If order does not matter: Combination.
• Examples:
  • Social Security numbers: Order matters (permutation).
  • Five-card hands in poker: Order does not matter (combination).
• There are always more permutations than combinations for the same $n$ and $k$.

Formulas for Permutations and Combinations

$<em>{n}P</em>{k} = n \times (n-1) \times (n-2) \times …$ (k factors)

• (Start with n and decrease by 1 until you have k factors).
• Example: Baseball lineup (9 players from 20).
  $<em>{20}P</em>{9} = 20 \times 19 \times 18 \times 17 \times 16 \times 15 \times 14 \times 13 \times 12$

$<em>{n}C</em>{k} = \frac{<em>{n}P</em>{k}}{k!} = \frac{n \times (n-1) \times (n-2) \times … }{k!}$

• Start like a permutation, but also divide by $k!$
• Example: 6 choose 3

$<em>{6}C</em>{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20$

• The result should always be a whole number.

Soccer Team Example

• Problem: Choosing 11 players for the field from 15 girls.
• Order does not matter (combination).

$<em>{15}C</em>{11} = \frac{15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5}{11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}$

• Simplify by canceling common factors.
• After canceling:
  $<em>{15}C</em>{11} = \frac{15 \times 7 \times 13}{1} = 1365$

Key Takeaways

• Permutations: Order matters (ordered subset).
• Combinations: Order does not matter (unordered subset).
• Use the appropriate formula.
• Simplify calculations by canceling common factors.