In this chapter, you will learn
- —Understand experimental and theoretical probability
- —Identify sample space and favorable outcomes
- —Apply probability in real-world scenarios
- —Calculate probabilities of complementary events
- —Solve compound probability problems
- —Understand limitations and applications of probability
Experimental Probability
Experimental probability is based on actual experiments or observations.
Formula: P(E) = Number of favorable outcomes / Total number of trials
Example: If a coin is flipped 100 times and heads appears 48 times, then experimental probability of heads = 48/100 = 0.48
Key Point: Experimental probability changes with more trials and approaches theoretical probability as trials increase (Law of Large Numbers)
Exam Tip
Be careful to distinguish between experimental probability (based on actual data) and theoretical probability (based on equally likely outcomes)
Common Mistake
Confusing experimental probability with theoretical probability. Experimental probability varies based on the number of trials performed.
Theoretical Probability
Theoretical probability is based on mathematical reasoning, assuming all outcomes are equally likely.
Formula: P(E) = Number of favorable outcomes / Total number of possible outcomes
Example: For a fair die, P(rolling a 3) = 1/6
Conditions: All outcomes must be equally likely for this formula to apply.
Exam Tip
Always check if outcomes are equally likely before using the theoretical probability formula
Common Mistake
Using theoretical probability when outcomes are not equally likely, or not counting all possible outcomes correctly
Sample Space and Events
The sample space is the set of all possible outcomes of an experiment.
An event is a subset of the sample space.
Example: For rolling a die, sample space S = {1, 2, 3, 4, 5, 6}
Event E (getting even number) = {2, 4, 6}
The number of elements in sample space is denoted as n(S), and number of favorable outcomes as n(E).
Exam Tip
Always clearly identify and list the sample space before solving probability problems
Common Mistake
Forgetting to consider all possible outcomes or listing the sample space incorrectly
Complementary Events
Two events E and E' are complementary if they are mutually exclusive and exhaustive.
Key Property: P(E) + P(E') = 1, or P(E') = 1 - P(E)
Example: If P(rain tomorrow) = 0.6, then P(no rain tomorrow) = 1 - 0.6 = 0.4
Complementary events cover all possible outcomes with no overlap.
Exam Tip
Use complementary events to simplify calculations - sometimes finding P(E') is easier than P(E)
Common Mistake
Confusing complementary events with mutually exclusive events. Complementary events must be mutually exclusive AND exhaustive.
Probability of Combined Events
For mutually exclusive events: P(A or B) = P(A) + P(B)
For independent events: P(A and B) = P(A) × P(B)
Example: P(rolling 2 or 5 on a die) = P(2) + P(5) = 1/6 + 1/6 = 2/6 = 1/3
Example: P(heads on coin AND 4 on die) = 1/2 × 1/6 = 1/12
Exam Tip
Identify whether events are mutually exclusive or independent before applying formulas
Common Mistake
Using multiplication rule for mutually exclusive events, or addition rule for independent events
Practical Applications
Probability is applied in weather forecasting, medical testing, gambling, insurance, quality control, and decision-making.
Weather: Probability of rainfall helps in planning
Medicine: Probability of side effects and treatment success
Quality Control: Probability of defective items in production
Limitations: Past results don't guarantee future outcomes. Each trial is independent (for fair experiments).
Exam Tip
Questions often involve real-world scenarios. Always identify the sample space clearly in word problems
Common Mistake
Assuming past outcomes affect future probability (gambler's fallacy)