More about Probability
Master conditional probability, addition laws, and multi-stage tree diagrams. Learn how to construct mathematical proofs of independence and avoid costly careless errors.
1. The Core Concept
Probability as Ratios: At its core, probability is about comparing favorable outcomes to total outcomes. However, in advanced scenarios, the "total outcomes" (sample space) changes dynamically — either because an event has already occurred (Conditional Probability) or because objects are removed from the system (Without Replacement).
2. DSE More about Probability Cheat Sheet — 5 Key Topics
Scan this compact reference table to review every essential concept, step, and shortcut in under 10 seconds.
| Topic | Key Concept & Steps | Exam Shortcuts & Tips |
|---|---|---|
| 1. Addition Law | P(A ∪ B) = P(A) + P(B) − P(A ∩ B) | • If A and B are mutually exclusive:P(A ∩ B) = 0 ⟹ P(A ∪ B) = P(A) + P(B) |
| 2. Conditional Probability | P(A | B) = P(A ∩ B) / P(B) | • Read as: "Probability of A given B has occurred." • The denominator shrinks to only include the outcomes inside event B. |
| 3. Independence of Events | A and B are independent if and only if:P(A ∩ B) = P(A) × P(B)(equivalently, P(A | B) = P(A)) | Proof Requirement: You must calculate the left-hand side and right-hand side separately, then write whether they are equal. |
| 4. Tree Diagrams (AP/GP) | • Along branches (AND): Multiply. • Across branches (OR): Add. | Used for multi-stage selection. Perfect for tracking "without replacement" games where probabilities change. |
| 5. P&C Probability | P(Event) = Favorable C(n,r) or P(n,r) / Total C(n,r) or P(n,r) | Used for drawing balls, cards, or forming committees where selection order does or does not matter. |
💡 Exam Traps & Secrets
⚠️ Conditional P(A|B) vs. Intersection P(A∩B) TrapThe difference:
P(A ∩ B) calculates the probability of both A and B occurring out of the entire original group. P(A | B) restricts the world to only group B, and calculates the probability of A occurring within that group.⚠️ The Independence Proof Trap (Paper 1 Step Marks)Never assume two events are independent just because they "sound" independent in the question text. You must prove it algebraically. How to write your steps:
• Calculate
• Calculate
• Compare: "Since LHS = RHS (or LHS ≠ RHS), the events A and B are independent (or dependent)."
• Calculate
LHS = P(A ∩ B)• Calculate
RHS = P(A) × P(B)• Compare: "Since LHS = RHS (or LHS ≠ RHS), the events A and B are independent (or dependent)."
⚠️ "Without Replacement" Denominator DecrementIn multi-stage drawing questions, if items are drawn successively without replacement, you must remember to decrease both your numerator and denominator by 1 for the subsequent branches. Example: Drawing 2 red balls from a bag of 4 red and 6 blue (total 10):
P(Both Red) = 4/10 × 3/9 = 12/90 = 2/15✏️ Self-Test: Quick Interactive Practice
How to study: Click any question below to instantly load it on the coordinate grid and check your steps.
(Tree diagram calculations with changing denominators)A bag has 4 red and 6 blue balls. Two balls are drawn successively without replacement. Find
P(both are red) and P(at least one is red).(Addition law, conditional probability, and independence proof steps)Given
P(A) = 0.4, P(B) = 0.5, and P(A ∩ B) = 0.2. Find P(A ∪ B) and P(A | B). Are A and B independent?(Combination-based probability calculation)From a class of 20 students (12 boys and 8 girls), a committee of 3 is chosen. Find the probability that there are at least 2 girls in the committee.