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
- Interpret Probabilities for Events
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
| Concept | Definition | Example |
|---|---|---|
| Sample Space (S) | All possible outcomes of a chance experiment | Rolling a die: {1,2,3,4,5,6} |
| Event | Subset of the sample space | Rolling an even number: {2,4,6} |
| Theoretical Probability | Based on equally likely outcomes | P(rolling a 3) = 1/6 |
| Experimental Probability | Based on observed outcomes from trials | 18 threes in 100 rolls → 0.18 |
| Subjective Probability | Based on personal judgment | 70% 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.
Interpret Probabilities for Events
Interpret Probabilities for Events
Probabilities of events in repeatable situations can be interpreted as the relative frequency with which the event occurs in the long run.
- Probability describes the proportion of times an outcome would occur if a random process were repeated many times under identical conditions.
- It does not predict what will happen in one trial but what we expect over many trials.
- As the number of trials increases, the relative frequency (observed probability) approaches the true theoretical probability.
Formal Definition:
If an event \( A \) occurs \( k \) times out of \( n \) trials, then the probability is approximated by:
\( P(A) \approx \dfrac{k}{n} \)
As \( n \to \infty \), the relative frequency \( \dfrac{k}{n} \) approaches the true probability \( P(A) \).
Interpretation in Context:
- \( P(A) = 0.2 \): In the long run, the event \( A \) occurs about 20% of the time.
- \( P(A) = 0.5 \): The event occurs about half the time in many repetitions.
- \( P(A) = 0.9 \): The event occurs almost every time (very likely).
- \( P(A) = 0.01 \): The event is very rare — it occurs about once in 100 trials on average.
Example :
A factory produces batteries, and the probability that a battery is defective is \( P(\text{defective}) = 0.05 \).
Interpret this probability.
▶️ Answer / Explanation
- If the production process continues under the same conditions for many batteries, about 5% of them will be defective in the long run.
- This means that in a batch of 1,000 batteries, we expect roughly \( 0.05 \times 1000 = 50 \) defective batteries.
- The probability does not guarantee exactly 5% in any one batch, but rather describes the long-run proportion over many batches.
Conclusion: Probability describes a long-run relative frequency, not a single short-term prediction.
Example :
A fair coin is flipped repeatedly. The theoretical probability of getting heads is \( P(\text{Heads}) = 0.5 \).
After 10 flips, the result was: 7 heads and 3 tails.
Question: Does this result contradict the probability model?
▶️ Answer / Explanation
- No — it does not contradict the model.
- In the short run, random variation can cause the proportion of heads to differ from 0.5.
- However, if the coin were flipped many more times (e.g., 1,000 or 10,000 flips), the relative frequency of heads would get closer to 0.5.
- This illustrates the Law of Large Numbers: the observed probability approaches the true probability as the number of trials increases.
Conclusion: \( P(\text{Heads}) = 0.5 \) means that in the long run, about half of all flips result in heads.