Basic Probability Lecture

Key Concepts in Probability

Sample Space

• Definition: All possible outcomes in an event.
• Example: A spinner with 7 numbers has a sample space labeled with all 7 numbers.
• Importance: Understanding sample space is crucial in statistics and higher probabilities (e.g., law of large numbers).
• Real-world Considerations: Sample space can significantly alter probability outcomes and is often manipulated in news stories for specific narratives.

Probability of an Event

• Example with Spinner: Probability of getting an odd number = 4 out of 7.
• Forms: Probability can be expressed as fractions or percentages.
• **Types of Probability: Theoretical vs. Experimental:
  • Theoretical Probability: Ratio of favorable outcomes to possible outcomes (e.g., coin toss = 1/2).
  • Experimental Probability: Ratio of actual outcomes in an experiment to the total trials (e.g., free throw success rate).**

Applications and Examples:

• Sports: Free throw percentages based on historical performance (experimental probability).
• Manufacturing: Error percentages monitored through experiments.

Theoretical vs Experimental Probability

Example with a Six-Sided Die

• **Theoretical:
  • Rolling a four = 1/6.
  • Rolling an odd number = 3/6 or 1/2.
• **Experimental:
  • Based on actual rolls, e.g., Rolling four 3 times out of 10 trials = 3/10.
• Law of Large Numbers: More trials bring experimental probability closer to theoretical probability.

Probability Calculations and Terms

Probability Range

• Range: Between 0 (impossible) and 1 (certain).
• Example Calculation: Flipping a coin: 1/2 for heads or tails.

Complement of an Event

• Definition: Opposite outcome of an event.
• Example: If probability of heads = 1/2, the complement (tails) = 1/2.
• Calculation: Probability of event + complement = 1.

Dependent and Independent Events

• Independent: Events that do not affect each other (flipping two different coins).
• Dependent: Second event probability changes based on the first event's outcome (drawing without replacement).

Mutually Exclusive vs Inclusive Events

• Mutually Exclusive: Events that cannot happen simultaneously (e.g., taking pre-calculus or calculus).
• Mutually Inclusive: Events that can happen together (e.g., students who play both baseball and basketball).
• Calculation: For inclusive events, subtract crossover probabilities.

Examples from Problem Solving

Survey and Calculation Example

• Question: 371 teens, 266 own dogs, 157 own cats, and 108 own both. Calculate probability for owning either.
• Calculation: P(dog) + P(cat) - P(dog and cat).
• Result: Approximately 85%.

Book Reading Survey Example

• **Survey of 434 students for books read over summer.
• Task: Probability of reading 4 or fewer books.
• Solution: Sum probabilities for reading 0, 1-2, or 3-4 books. Result = ~37.8%.
• Task for 5 books or more: Use complement rule (1 - P(four or fewer)).**

Final Key Points

• Events: Always consider event dependencies and mutual exclusivity.
• Visualization: Use Venn diagrams to visualize and solve probability problems.