lesson 12 continued

Probability Basics

• Definition: Probability measures the likelihood of a specific event occurring, ranging from 0 (impossible) to 1 (certain).

Calculating Probability of Correct Answers

• Example: If guessing on a test with 4 options per question:
  • Correct answer probability: $P(C) = \frac{1}{4} = 0.25$
  • Incorrect answer probability: $P(I) = 1 - P(C) = \frac{3}{4} = 0.75$

Multiple Questions Scenario

• For answering 5 questions incorrectly:

  • Probability of getting all incorrect answers:
    $P(I) = (0.75)^5 = 0.2373$ (approximately)

• To find the probability of getting at least one correct answer:

  • Use the complement rule:
    $P(at\ least\ one\ correct) = 1 - P(all\ incorrect)$
    $= 1 - 0.2373 = 0.7627$ (approximately)

Straightforward Calculation

• Alternatively, you can find the probability of getting one, two, or more correct answers directly, but this is often complex because it requires tracking multiple outcomes.

General Probability Rules

• Multiplication Rule:
  • If events A and B are independent,
    $P(A \text{ and } B) = P(A) \times P(B)$

Conditional Probability

• Definition:
  • Conditional probability measures the probability of event B occurring given that event A has occurred.
  • Notation:
    $P(B | A)$
  • Formally:
    $P(B | A) = \frac{P(A \text{ and } B)}{P(A)}$
  • Example:
  • Probability of a person being pregnant given a positive test result.
    • Calculate based on the relevant portion of a given data set (e.g., a table).

Practice Problems on Conditional Probability

• Working through examples with a provided table covering test results (pregnant/not pregnant, positive/negative).

• Key idea is to focus only on the given condition when performing calculations (e.g., focus on rows/columns related to positive test results).

General Multiplication Rule

• For two events A and B:
  • 
    $P(A \text{ and } B) = P(A) \times P(B | A)$
• This allows addressing scenarios where choosing one outcome affects the outcome of another.

Independence of Events

• To check if two events A and B are independent:
  • If
    $P(B | A) = P(B)$
    then events are independent.

Practical Examples

• Discussed examples included scenarios like machine failures or outcomes of marble picking.
• Example of picking marbles without replacement illustrates how probabilities change as each event alters the sample size.

Alternate Calculating Method for Conditional Probability

• If having trouble calculating $P(B | A)$ directly:
  • Use the formula:
    $P(B | A) = \frac{P(A \text{ and } B)}{P(A)}$
  • This method simplifies some tasks by focusing on known event pairs directly from historical data or a table.

Conclusion and Further Topics

• Move toward counting methods and their applications in probability questions and scenarios.