Probability and Counting Techniques

How Many Ways (HMW)

• HMW will be used to represent "How Many Ways" in problems.

Sample Space and Tree Diagrams

• A sample space lists every possible outcome.
• A tree diagram is a visual representation of a sample space, useful for illustrating initial concepts but can become bulky.
• Example: Nick's Outfit Combinations (Doors, Top, Body)
  • Decisions to make: color of doors, color of top, color of body.
  • Possible door colors: red, white, blue (3 choices).
  • Top colors: must be different from the door color (2 choices).
  • Body colors: green, yellow (2 choices).
  • A complete tree diagram will show all possible combinations.
  • Total outcomes are found by counting the branches at the end of the tree. In this case, there are 12 total outcomes.

Fundamental Counting Principle (Slot Method)

• Also known as the slot method.
• If two events are done in succession: the first event can be done in $x$ ways, and the second event can be done in $y$ ways, then the total number of ways that can be done is equal to $x \times y$.
• This can be extended to multiple events.

Applying the Slot Method to Nick's Outfit

• Three events: doors, top, body.
• Slots representing each event.
  • Doors: 3 choices.
  • Top: 2 choices (different from door color).
  • Body: 2 choices.
• Total number of ways: $3 \times 2 \times 2 = 12$.
• If between events, the word "and" makes sense, this means multiply.

License Plate Problem

• License plates consist of three letters followed by four numbers.

No Restrictions

• 26 letters in the alphabet.
• 10 digits (0-9).
• Number of ways: $26 \times 26 \times 26 \times 10 \times 10 \times 10 \times 10 = 26^3 \times 10^4 = 175,760,000$.
• Multiply because it's a letter AND a letter AND a letter and a number AND a number AND a number.

No Repetition Allowed

• Number of ways: $26 \times 25 \times 24 \times 10 \times 9 \times 8 \times 7 = 32,760,000$.
• If letters can repeat, but numbers can't: $26 \times 26 \times 26 \times 10 \times 9 \times 8 \times 7$.

Seating Arrangement Problem

• Six people to be seated in a row of six seats.

No Restrictions

• Choices for each seat: 6, 5, 4, 3, 2, 1.
• Total number of ways: $6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$.
• This is also 6 factorial (6!).
• $n!$ (n factorial) means multiply every number from n down to 1.
• Calculators have a factorial function.

Couples Must Sit Together

• Assume couples are boy/girl pairs.
• Choices: 6, 1, 4, 1, 2, 1 (the "1" represents the date that has to sit with the person).
• Total ways: $6 \times 1 \times 4 \times 1 \times 2 \times 1 = 48$.

Couples Sit Together on the Left

• Choices: 3, 1, 2, 1, 1, 1 (three options for the first couple -- first two seats).
• Total ways: $3 \times 1 \times 2 \times 1 \times 1 \times 1 = 6$.

Five-Digit Number Problems

• A five-digit number cannot start with zero.

Digits Cannot Repeat

• First digit: 9 choices (1-9).
• Second digit: 9 choices (can include 0, but not the first digit).
• Remaining digits: 8, 7, 6.
• Total numbers: $9 \times 9 \times 8 \times 7 \times 6 = 27,216$.

Adjacent Digits Must Be Different

• First digit: 9 choices (1-9).
• Second digit: 9 choices (can be 0, but not the first digit).
• Remaining digits: 9 choices (cannot be the same as the adjacent digit).
• Total numbers: $9 \times 9 \times 9 \times 9 \times 9 = 9^5 = 59,049$.

No Repeats, Number is Odd

• Last digit must be odd (1, 3, 5, 7, 9) - 5 choices.
• First digit: 8 choices (cannot be 0 or the last digit).
• Second digit: 8 choices (can be 0, but not the first or last digit).
• Third digit: 7 choices.
• Fourth digit: 6 choices.
• Total numbers: $8 \times 8 \times 7 \times 6 \times 5 = 13,440$.

Sample Spaces for Kids or Coin Flips

• Specifically designed for scenarios with two outcomes (e.g., boy/girl, heads/tails).

Three Kids, Exactly Two Girls

• Three events (births), two outcomes for each.
• Total possible outcomes: $2 \times 2 \times 2 = 8$.
• Sample Space:
  • Start with total number of outcomes and divide by 2. 8/2 = 4
  • List four boys, then four girls: BBBB GGGG
  • Halve it again: two boys, two girls: BB GG BB GG
  • Halve again: alternate boy and girl: BGBGBGBG
• The sample space has all unique outcomes.
• Ways to get two girls: BBG, BGB, GBB (3 ways).

Flip Four Coins, Get Three Tails

• Four events two outcomes for each.
• Total possible outcomes: $2 \times 2 \times 2 \times 2 = 16$.
• Sample Space:
  • Start with total number of outcomes and divide by 2. 16/2 = 8
  • List eight heads, then eight tails: HHHHHHHH TTTTTTTT
  • Halve it again: four heads, then four tails: HHHH TTTT HHHH TTTT
  • Halve again: two heads, then two tails: HHTT HHTT HHTT HHTT
  • Halve again: alternate head and tail: HTHTHTHTHTHTHTHT
• Looking for three tails: HTTT, THTT, TTHT, TTTH (4 ways).

Sample Space for Rolling Two Dice

• Outcomes range from 1-6 on each die.
• Sample space is all the combinations of dice rolls.

Creating the Sample Space

• (1,1), (1,2), (1,3), (1,4), (1,5), (1,6)
• (2,1), (2,2), (2,3), (2,4), (2,5), (2,6)
• (3,1), (3,2), (3,3), (3,4), (3,5), (3,6)
• (4,1), (4,2), (4,3), (4,4), (4,5), (4,6)
• (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)
• (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)

Using the Sample Space

• Ways to get a sum equal to 6: (1,5), (2,4), (3,3), (4,2), (5,1) (5 ways).
• Diagonals from bottom left to top right have the same sum.
• Ways to get a sum less than 7: count all combinations that sum to less than 7 (15 ways).
• Ways to get a sum equal to 4 or 9: (1,3), (2,2), (3,1) and (3,6), (4,5), (5,4), (6,3) (7 ways).

Five on Either Die

• Count combinations with a 5 on either die, but don't double-count (11 ways).
• The word "or" means to include possibilities from either event.