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AP Statistics 4.3 Introduction  to Probability Study Notes

AP Statistics 4.3 Introduction  to Probability Study Notes- New syllabus

AP Statistics 4.3 Introduction  to Probability Study Notes -As per latest AP Statistics Syllabus.

LEARNING OBJECTIVE

  • The likelihood of a random event can be quantified.

Key Concepts:

  • Introduction to Probability

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Introduction to Probability

Introduction to Probability

Probability is the study of randomness. It provides a mathematical framework for describing uncertain events and predicting long-term outcomes. In AP Statistics, probability connects random processes to statistical inference.

Key Concepts of Probability

  • Randomness: Outcomes are uncertain in the short run but follow predictable patterns in the long run.
  • Experiment: A repeatable process that produces outcomes (e.g., rolling a die, flipping a coin).
  • Outcome: A possible result of a chance experiment (e.g., rolling a “4”).
  • Sample Space (S): The set of all possible outcomes (e.g., S = {1,2,3,4,5,6} for a die roll).
  • Event: A subset of the sample space (e.g., rolling an even number = {2,4,6}).

Rules of Probability

  • Probabilities are always between 0 and 1 (0 ≤ P(A) ≤ 1).
  • The probability of the entire sample space is 1: P(S) = 1.
  • The probability of an event not happening: P(Not A) = 1 − P(A).
  • If two events are mutually exclusive (cannot occur together), then P(A or B) = P(A) + P(B).
  • In general: P(A or B) = P(A) + P(B) − P(A and B).

Types of Probability

  • Theoretical Probability: Based on mathematical reasoning and equally likely outcomes (e.g., P(rolling a 3) = 1/6).
  • Experimental Probability: Based on actual trials and outcomes (e.g., rolling a die 100 times and getting 18 threes → 18/100 = 0.18).
  • Subjective Probability: Based on personal judgment or experience, not precise calculations (e.g., “I think there’s a 70% chance it will rain tomorrow”).

Comparison Table

ConceptDefinitionExample
Sample Space (S)All possible outcomes of a chance experimentRolling a die: {1,2,3,4,5,6}
EventSubset of the sample spaceRolling an even number: {2,4,6}
Theoretical ProbabilityBased on equally likely outcomesP(rolling a 3) = 1/6
Experimental ProbabilityBased on observed outcomes from trials18 threes in 100 rolls → 0.18
Subjective ProbabilityBased on personal judgment70% chance of rain

Example 

What is the probability of rolling a number greater than 4 on a fair six-sided die?

▶️ Answer / Explanation

Step 1: Sample space = {1,2,3,4,5,6}.

Step 2: Favorable outcomes = {5,6} → 2 outcomes.

Step 3: Probability = 2/6 = 1/3.

Conclusion: P(number > 4) = 1/3.

Example 

A spinner has 4 equal sections labeled A, B, C, and D. What is the probability of not landing on C?

▶️ Answer / Explanation

Step 1: Sample space = {A, B, C, D}.

Step 2: P(C) = 1/4.

Step 3: P(Not C) = 1 − 1/4 = 3/4.

Conclusion: Probability = 0.75.

Example 

In 50 rolls of a die, a student gets “2” exactly 11 times. What is the experimental probability of rolling a 2?

▶️ Answer / Explanation

Step 1: Number of favorable outcomes = 11.

Step 2: Total trials = 50.

Step 3: Experimental probability = 11/50 = 0.22.

Conclusion: P(rolling a 2) ≈ 0.22.

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