Foundational definitions and terminology essential for understanding probability theory in CA Foundation exams.
## Core concept
Probability is the mathematical measure of likelihood that an event will occur. In CA Foundation, you must master the language and foundational ideas before tackling theorems.
Sample Space (S): The set of all possible outcomes of a random experiment. - Example: Tossing a coin → S = {H, T} - Example: Rolling a die → S = {1, 2, 3, 4, 5, 6}
Event (A, B, etc.): Any subset of the sample space; a specific outcome or combination of outcomes. - Simple event: Single outcome (e.g., getting 6 on a die) - Compound event: Two or more outcomes (e.g., getting an even number on a die = {2, 4, 6})
Mutually Exclusive Events: Two events cannot happen simultaneously. - Rolling a die: getting 2 AND getting 5 are mutually exclusive - If A and B are mutually exclusive, A ∩ B = ∅
Exhaustive Events: All possible outcomes of an experiment are covered. - Coin toss: {H, T} is exhaustive - If events are exhaustive, their union equals the sample space: A ∪ B ∪ C = S
Independent Events: The occurrence of one event does not affect the probability of another. - Tossing two coins: result of first toss does not change probability of second toss - Drawing with replacement: drawing a card, replacing it, then drawing again
Dependent Events: The occurrence of one event *does* affect the probability of another. - Drawing without replacement: removing one card changes the composition of the deck for the next draw
## Terminology and notation
| Term | Symbol | Meaning | |---|---|---| | Event A | A | Specific outcome or set of outcomes | | Complement of A | A' or A^c | All outcomes NOT in A | | A and B occur | A ∩ B | Intersection; both events happen | | A or B occur | A ∪ B | Union; at least one event happens | | Probability of A | P(A) | Likelihood of A; always 0 ≤ P(A) ≤ 1 | | Conditional probability | P(A\|B) | Probability of A given B has occurred |
## Fundamental rules
- Certain event: P(S) = 1 (something in the sample space will happen)
- Impossible event: P(∅) = 0 (nothing cannot happen)
- Complement rule: P(A) + P(A') = 1
- Number of outcomes: If all outcomes are equally likely, P(A) = (Number of favorable outcomes) / (Total number of outcomes)
## Common exam applications
- Identifying the sample space: A die is rolled twice. Sample space has 6 × 6 = 36 outcomes.
- Checking mutual exclusivity: "A customer buys tea" and "buys coffee" may NOT be mutually exclusive (customer could buy both). Confirm based on problem context.
- Finding complements: If P(defective item) = 0.05, then P(non-defective) = 0.95.
## Worked example
Question: A bag contains 3 red balls and 5 blue balls. Two balls are drawn *without replacement*. Define the sample space and identify whether the events "first ball is red" and "second ball is blue" are dependent or independent.
Solution: - Total balls = 8 - Sample space S = {all possible pairs of 2 balls from 8} = 28 outcomes (combinations) - Event A: "First ball is red" → 3 × 7 = 21 outcomes (3 red balls for first draw, 7 remaining for second) - Event B: "Second ball is blue" → depends on what was drawn first - Since we draw *without replacement*, the composition of the bag changes after the first draw. Therefore, A and B are dependent events. - If drawn *with replacement*, they would be independent.
## Common mistakes
- Confusing "mutually exclusive" with "independent" — mutually exclusive events *cannot* be independent (if one happens, the other cannot).
- Forgetting that probabilities must sum to 1 when all possibilities are listed.
- Not reading carefully whether sampling is "with" or "without" replacement; this determines independence.