How to Solve: IB AI HL/SL – Probability (Tree Diagrams, Venn, Markov Chains, Bayes)

How to Solve: IB AI HL/SL – Probability (Tree Diagrams, Venn, Markov Chains, Bayes)

Introduction

"Mastering probability in IB Math AI HL/SL can add 10–15% to your exam score—because it’s the backbone of Paper 2 questions on risk, medical testing, and even AI decision-making. One tree diagram or Bayes’ Theorem question can be the difference between a 5 and a 7."

What You Need To Know First

1. Basic probability rules: ( P(A) + P(A') = 1 ), ( P(A \cup B) = P(A) + P(B) - P(A \cap B) ).
2. Conditional probability: ( P(A|B) = \frac{P(A \cap B)}{P(B)} ).
3. Independent vs. dependent events: Independent if ( P(A \cap B) = P(A) \times P(B) ).

Key Vocabulary

Formulas To Know

1. Tree Diagrams

• Multiplication rule (AND): ( P(A \text{ and } B) = P(A) \times P(B|A) ).
• Addition rule (OR): ( P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) ).
• MEMORISE THIS: For mutually exclusive events, ( P(A \text{ or } B) = P(A) + P(B) ).

2. Venn Diagrams

• Union: ( P(A \cup B) = P(A) + P(B) - P(A \cap B) ).
• MEMORISE THIS: If ( A ) and ( B ) are mutually exclusive, ( P(A \cap B) = 0 ).
• Complement: ( P(A') = 1 - P(A) ).

3. Bayes’ Theorem

[ P(A|B) = \frac{P(B|A) \times P(A)}{P(B)} ] - ( P(A) ): Prior probability. - ( P(B|A) ): Likelihood. - ( P(B) ): Total probability (use law of total probability if needed). - MEMORISE THIS: Often given on exam sheet, but know how to apply it.

4. Markov Chains

• Transition matrix (P): Probabilities of moving from one state to another.
• Initial state vector (v₀): Starting probabilities.
• Next state: ( v_1 = v_0 \times P ).
• MEMORISE THIS: Matrix multiplication rules apply.

Step-by-Step Method

Tree Diagrams

1. Draw branches for each stage (e.g., first event, second event).
2. Label probabilities on each branch (use decimals or fractions).
3. Multiply along branches for "AND" probabilities.
4. Add probabilities for "OR" scenarios (mutually exclusive).
5. Check: All probabilities at a node must sum to 1.

Venn Diagrams

1. Draw circles for each event (label them).
2. Fill in intersections first (e.g., ( P(A \cap B) )).
3. Work outward: Subtract intersections from individual probabilities.
4. Shade regions for ( P(A \cup B) ), ( P(A') ), etc.
5. Verify: Total probability inside the rectangle = 1.

Bayes’ Theorem

1. Identify given probabilities: Prior (( P(A) )), likelihood (( P(B|A) )), and evidence (( P(B) )).
2. Calculate ( P(B) ) using law of total probability if needed:
   [ P(B) = P(B|A) \times P(A) + P(B|A') \times P(A') ]
3. Plug into Bayes’ formula:
   [ P(A|B) = \frac{P(B|A) \times P(A)}{P(B)} ]
4. Simplify and interpret the result.

Markov Chains

1. Define states (e.g., "Sunny," "Rainy").
2. Write transition matrix (P): Rows = current state, columns = next state.
3. Multiply initial vector (v₀) by P to get next state (v₁).
4. Repeat multiplication for future states (v₂ = v₁ × P).
5. Find steady-state (if asked): Solve ( v = v \times P ) and ( \sum v = 1 ).

Worked Examples

Example 1 – Basic Tree Diagram

Question: A bag has 3 red and 2 blue marbles. Two marbles are drawn without replacement. Find ( P(\text{one red and one blue}) ).

Steps: 1. Draw first branch: Red (3/5) or Blue (2/5). 2. From Red, second branch: Red (2/4) or Blue (2/4). 3. From Blue, second branch: Red (3/4) or Blue (1/4). 4. Multiply for "Red then Blue": ( \frac{3}{5} \times \frac{2}{4} = \frac{6}{20} ). 5. Multiply for "Blue then Red": ( \frac{2}{5} \times \frac{3}{4} = \frac{6}{20} ). 6. Add: ( \frac{6}{20} + \frac{6}{20} = \frac{12}{20} = \frac{3}{5} ).

What we did and why: We used the tree to visualize all paths, multiplied for "AND," and added for "OR."

Example 2 – Medium Venn Diagram

Question: In a class, 60% like math, 50% like physics, and 30% like both. Find ( P(\text{math only}) ).

Steps: 1. Draw two intersecting circles: Math (M) and Physics (P). 2. Fill ( P(M \cap P) = 0.3 ). 3. ( P(M \text{ only}) = P(M) - P(M \cap P) = 0.6 - 0.3 = 0.3 ). 4. Verify: ( P(P \text{ only}) = 0.5 - 0.3 = 0.2 ). 5. Total: ( 0.3 + 0.2 + 0.3 = 0.8 ) (remaining 20% like neither).

What we did and why: We isolated "math only" by subtracting the overlap from the total math probability.

Example 3 – Exam-Style Bayes’ Theorem

Question: A disease affects 1% of a population. A test is 95% accurate (5% false positives). If a person tests positive, what’s the probability they have the disease?

Steps: 1. Define:
- ( P(D) = 0.01 ) (prior).
- ( P(T|D) = 0.95 ) (true positive).
- ( P(T|D') = 0.05 ) (false positive). 2. Calculate ( P(T) ):
[ P(T) = P(T|D) \times P(D) + P(T|D') \times P(D') = 0.95 \times 0.01 + 0.05 \times 0.99 = 0.059 ] 3. Apply Bayes’:
[ P(D|T) = \frac{0.95 \times 0.01}{0.059} \approx 0.161 ] 4. Answer: ~16.1%.

What we did and why: We used Bayes’ to update the prior probability with new evidence (the test result).

Common Mistakes

1. Mistake: Adding probabilities for non-mutually exclusive events without subtracting the intersection.
   WHY IT HAPPENS: Forgetting the ( P(A \cap B) ) term in ( P(A \cup B) ).
   CORRECT APPROACH: Always use ( P(A \cup B) = P(A) + P(B) - P(A \cap B) ).

2. Mistake: Multiplying probabilities for dependent events as if they’re independent.
   WHY IT HAPPENS: Ignoring "without replacement" scenarios.
   CORRECT APPROACH: Use ( P(B|A) ) for dependent events.

3. Mistake: Mislabeling tree diagram branches (e.g., putting probabilities on the wrong level).
   WHY IT HAPPENS: Rushing the diagram.
   CORRECT APPROACH: Label each branch clearly and check sums at nodes.

4. Mistake: Forgetting to normalize probabilities in Markov chains (e.g., not ensuring ( \sum v = 1 )).
   WHY IT HAPPENS: Skipping the final step of dividing by the sum.
   CORRECT APPROACH: Always verify probabilities sum to 1.

5. Mistake: Confusing ( P(A|B) ) with ( P(B|A) ) in Bayes’ Theorem.
   WHY IT HAPPENS: Mixing up the order of conditions.
   CORRECT APPROACH: Write down what’s given (e.g., "test positive given disease" vs. "disease given test positive").

Exam Traps

1. Trap: Giving probabilities as percentages but expecting answers as decimals.
   How to Spot it: Question says "20% probability" but answer space is blank (no % sign).
   How to Avoid it: Convert all percentages to decimals (e.g., 20% → 0.2) before calculations.

2. Trap: Markov chain questions with "steady-state" but no initial vector.
   How to Spot it: Asks for "long-term probabilities" without giving starting values.
   How to Avoid it: Solve ( v = v \times P ) and ( \sum v = 1 ) (set up equations and solve).

3. Trap: Bayes’ Theorem questions where ( P(B) ) isn’t directly given.
   How to Spot it: Only provides ( P(B|A) ) and ( P(B|A') ), not ( P(B) ).
   How to Avoid it: Use the law of total probability to find ( P(B) ).

1-Minute Recap

"Here’s your last-minute cheat sheet for IB Probability: 1. Tree diagrams: Multiply along branches for ‘AND,’ add for ‘OR.’ 2. Venn diagrams: Fill intersections first, then subtract for ‘only’ regions. 3. Bayes’ Theorem: Prior × Likelihood / Evidence. If ( P(B) ) is missing, use total probability. 4. Markov chains: Multiply initial vector by transition matrix. For steady-state, solve ( v = v \times P ). 5. Common traps: Convert percentages to decimals, don’t confuse ( P(A|B) ) with ( P(B|A) ), and always check if events are independent. You’ve got this—now go ace that exam!