Lesson 12.1

12.1 THE NATURE OF PROBABILITY

Overview of Probability

  • Experiment: A controlled operation yielding a set of results.

  • Outcomes: The possible results of an experiment.

  • Event: A subcollection of outcomes from an experiment.

  • Example: A die has six faces, numbered 1 to 6. The sum of dots on opposite faces equals 7.

Types of Probability

  • Empirical Probability (Experimental)

    • Definition: Relative frequency of occurrence of an event based on actual experiment observations.

    • Notation: Indicated as P(E) which means "probability of event E".

  • Theoretical Probability (Mathematical)

    • Definition: Determined through studying the possible outcomes for a given experiment.

12.1.1 EMPIRICAL PROBABILITY (RELATIVE FREQUENCY)

  • Formula: P(E) = (Number of times event E has occurred) / (Total experiments performed).

Example 1: Coin Toss Probability

  • Scenario: In 100 tosses of a fair coin, 44 landed heads up.

  • Empirical Probability Calculation: P(Heads) = 44/100 = 0.44.

Example 2: Weight Reduction Drug Test

  • Context: A pharmaceutical company tests a weight reduction drug on 500 individuals.

    • Outcomes:

      • Weight Reduced: 379

      • Weight Unchanged: 62

      • Weight Increased: 59

  • Empirical Probability Calculations:

    • (a) P(Weight Reduced) = 379/500 = 0.758.

    • (b) P(Weight Unchanged) = 62/500 = 0.124.

    • (c) P(Weight Increased) = 59/500 = 0.118.

12.1.2 THE LAW OF LARGE NUMBERS

  • Concept: The law states that as the number of trials increases, the empirical probability tends to approach the theoretical probability.

  • Intuition: For a fair coin, the probability of landing heads up is expected to be 1/2.

  • Important Consideration:

    • Tossing a coin multiple times does not guarantee an equal split of heads and tails in a small number of trials.

Expected Vs Actual Outcomes

  • Table Summary:

    • Tosses | Expected Heads | Observed Heads | Relative Frequency (Heads)

      • 4 tosses: expected 2, observed 2, relative frequency = 0.5

      • 100 tosses: expected 50, observed 45, relative frequency = 0.45

      • 1000 tosses: expected 500, observed 546, relative frequency = 0.546

      • 10,000 tosses: expected 5000, observed 4852, relative frequency = 0.4852

      • 100,000 tosses: expected 50000, observed 49770, relative frequency = 0.49770

  • Conclusion: The relative frequency of heads approaches 0.5 as more tosses are performed, supporting the Law of Large Numbers.