Theoretical Probability, Permutations and Combinations

Understanding Probability

  • Probability is a mathematical concept that integrates into everyday decision-making.

  • People often base decisions on perceived probabilities of success and happiness.

  • While some probabilities (like personal happiness) are hard to calculate practically, basic probabilities are mathematically computable.

Basic Probability Concepts

  • Probability Definition: Probability of an event is calculated as the ratio of favorable outcomes to the total number of possible outcomes.

    • Example: Flipping a coin

      • Outcomes: Heads (H) and Tails (T)

      • Probability of Heads = 1 favorable outcome / 2 total outcomes = 1/2 or 0.5 (50%)

Multiple Events

  • Probability of predicting two heads when flipping a coin twice:

    • Outcomes: HH, HT, TH, TT (4 possible outcomes)

    • Probability for first flip (H) = 1/2, for second flip (H) = 1/2

    • Combined Probability = 1/2 * 1/2 = 1/4 (0.25 or 25%)

Special Cases of Probability

  • Impossible Event: Probability of 0 (event will not happen)

  • Certain Event: Probability of 1 (event will definitely happen)

Example with Socks

  • Socks in a drawer: 5 red, 3 blue, 7 green, 10 yellow (total 25 socks)

    • Probability of drawing a red sock = 5 red / 25 total = 0.2 (20%)

    • Probability of drawing two red socks:

      • First red sock: 5/25

      • Second red sock: 4/24

      • Combined Probability = (5/25) * (4/24) = 0.033 (3.3%)

Dice Probability

  • Rolling a pair of dice

    • Probability of getting a sum of 12 = 1 (only outcome: double sixes) / 36 total possible outcomes = 1/36

    • Probability of getting a sum of 8 or more:

      • Possible combinations for sums of 8, 9, 10, 11, and 12.

      • Total successful outcomes = 15 / 36 = 41.7% probability

Combination Locks

  • Calculating Combinations of a Lock:

    • 3 digits, each with 10 options = 10^3 possibilities = 1000 combinations

Permutation Examples

  • Race Scenario: Finding outcomes for top three finishers:

    • Winners can’t be reused, leading to possible outcomes for winner (10), second place (9), and third place (8).

    • Total outcomes = 10! / (10-3)!

Poker Hands and Combinations

  • Five-Card Poker Hands:

    • First card: 52 possibilities, second card: 51, third: 50, fourth: 49, fifth: 48.

    • Total combinations = 52!/(52-5)!

    • Adjust for order (permutations): divide by 5! to account for hand order having no significance.

    • Result: total combinations of five-card hands = over 2.5 million.

Specific Poker Hands Probability

  • Royal Flush: 1 per suit, 4 suits = 4 total. Resulting probability is minuscule.

  • Straight Flush: 9 possible combinations per suit x 4 suits = 36 total.

  • Four of a Kind: 13 possible combinations x 48 for additional card = 624 total.

  • Three of a Kind: 13 options x 4 combinations x 48 (fourth card options) x 44 (fifth card options) ÷ 2 = 54,912 total combinations. Average probability is slightly above 2%.

Combinations Summary

  • Understanding of permutations and combinations aids in calculating probabilities in various scenarios, from simple events to complex games like poker.