probability-basics - Obsidian Publish

# Working with Probabilities in Decision Trees #decision-tree #probability #estimation #calibration #uncertainty Master the art of probability estimation for decision trees. Learn reliable methods for assigning probabilities and dealing with uncertainty. ← [[node-types|Back to Node Types]] ## 🎲 Probability Fundamentals ### What Are We Estimating? In decision trees, probabilities represent: - **Likelihood of uncertain events occurring** - **Based on available information** - **Expressed as percentages (0-100%)** - **Must sum to 100% for each chance node** ### Why Probabilities Matter ``` Without Probabilities: Option A could result in $100K or $0 Option B guarantees $40K → Which is better? Unclear. With Probabilities: Option A: 30% chance of $100K, 70% chance of $0 = $30K expected Option B: 100% chance of $40K = $40K expected → Option B is better on average. ``` ## 📊 Probability Estimation Methods ### 1. Historical Data (Most Reliable) Use past data when available and relevant: #### Example: Job Application Success ``` Personal Track Record: - Applications sent: 50 - Interview invitations: 12 - Job offers: 3 Probability Estimates: Interview Rate: 12/50 = 24% Offer Rate: 3/12 = 25% Overall Success: 3/50 = 6% ``` #### Example: Product Launch Success ``` Company History: - Product launches: 8 - Major successes: 2 - Moderate successes: 4 - Failures: 2 Estimated Probabilities: Major Success: 2/8 = 25% Moderate Success: 4/8 = 50% Failure: 2/8 = 25% ``` #### Adjusting Historical Data Historical data may need adjustment: ``` Base Rate: 60% startup success rate Your Situation: First-time founder, limited funding Adjustment: Reduce to 40-45% Base Rate: 80% promotion rate in your company Your Situation: Top performer, strong relationships Adjustment: Increase to 90-95% ``` ### 2. Reference Class Forecasting Look at similar situations and outcomes: #### Example: MBA ROI ``` Reference Class: Working professionals pursuing MBA Data Source: Business school employment reports Career Advancement Probabilities: Significant promotion within 3 years: 70% Moderate advancement: 20% No change: 10% Salary Increase Probabilities: >50% increase: 25% 25-50% increase: 45% <25% increase: 30% ``` #### Example: Real Estate Investment ``` Reference Class: Similar properties in area Data Source: Local real estate reports 5-Year Appreciation Probabilities: High appreciation (>20% total): 30% Moderate appreciation (10-20%): 50% Low/no appreciation (<10%): 20% ``` ### 3. Expert Consultation Seek informed opinions from experienced individuals: #### Medical Decision Example ``` Consulting Physician Assessment: Treatment Success Rates: - Surgery: 85% success, 10% partial, 5% no improvement - Medication: 60% success, 30% partial, 10% no improvement - Physical therapy: 40% success, 50% partial, 10% no improvement Based on: 20 years experience, similar cases ``` #### Business Strategy Example ``` Industry Expert Opinion: Market Entry Probabilities: - Regulatory approval: 75% (based on recent trends) - Customer adoption: 60% (similar product history) - Competitive response: 80% (standard practice) Source: 15-year industry veteran, consulting firm data ``` ### 4. Structured Estimation Techniques #### The Delphis Method Multiple expert opinions with iterative refinement: ``` Round 1: 5 experts estimate independently Round 2: Share anonymous results, re-estimate Round 3: Discuss reasoning, final estimates Example Results: Expert estimates for market success: Round 1: 30%, 45%, 60%, 35%, 50% Round 2: 40%, 45%, 55%, 40%, 45% Round 3: Consensus around 40-45% ``` #### Three-Point Estimation Estimate optimistic, pessimistic, and most likely: ``` New Product Success: Optimistic (best case): 80% Most Likely (expected): 45% Pessimistic (worst case): 20% Weighted Average: (80 + 4×45 + 20) / 6 = 45% (Gives more weight to most likely scenario) ``` #### Confidence Intervals Express uncertainty about your estimates: ``` Market Share Capture: Point Estimate: 15% Confidence Range: 10-25% (80% confident) Wide range indicates high uncertainty ``` ## 🎯 Calibration Techniques ### Testing Your Probability Intuition #### Calibration Exercise Make 100 probability predictions and track accuracy: ``` Prediction: "80% chance it will rain tomorrow" Outcome: Track if it actually rains Goal: When you say 80%, it should happen ~80% of the time ``` #### Common Calibration Errors **Overconfidence Bias:** ``` ❌ Poor Calibration: "90% sure this will succeed" → Actually succeeds 60% of time ✅ Better Calibration: "60% sure this will succeed" → Actually succeeds 60% of time ``` **Binary Thinking:** ``` ❌ Poor: Only using 10%, 50%, 90% ✅ Better: Using full range 15%, 35%, 65%, 85% ``` ### Improving Probability Estimates #### 1. Outside View Step back from your specific situation: ``` Inside View: "My startup is special, 90% chance of success" Outside View: "Most startups fail, 10-20% typically succeed" Balanced Estimate: 15-25% chance of success ``` #### 2. Consider Alternative Scenarios ``` Question: What could go wrong? - Competitor launches first (20% probability) - Technology doesn't work (15% probability) - Market smaller than expected (25% probability) - Regulatory delays (10% probability) Adjust success probability downward based on risks ``` #### 3. Use Frequency Format ``` Natural Frequency: "Out of 100 similar situations, how many succeed?" Often easier than "What's the probability of success?" Example: "8 out of 10 similar products succeeded" = 80% probability of success ``` ## 📈 Advanced Probability Concepts ### 1. Conditional Probabilities Probabilities that depend on other events: #### Example: Business Success ``` IF economic conditions are good: Success probability: 70% IF economic conditions are poor: Success probability: 30% Current economic assessment: Good conditions: 60% probability Poor conditions: 40% probability Overall success probability: (0.6 × 0.7) + (0.4 × 0.3) = 0.42 + 0.12 = 54% ``` #### Example: Medical Treatment ``` IF patient is young and healthy: Treatment success: 90% IF patient is elderly with complications: Treatment success: 60% This patient profile matches: Young and healthy: 70% match Elderly with complications: 30% match Expected success rate: (0.7 × 0.9) + (0.3 × 0.6) = 63% + 18% = 81% ``` ### 2. Updating Probabilities (Bayesian Approach) Revise estimates as new information arrives: #### Initial Estimate ``` Product Success Probability: 60% Based on: Market research, competitor analysis ``` #### New Information Arrives ``` Beta test results: 85% user satisfaction Historical pattern: High satisfaction correlates with success Revised Probability: 75% Updated based on: Strong user feedback ``` #### Systematic Updating Process ``` Prior Probability: P(Success) = 60% New Evidence: Beta test positive Likelihood: P(Positive Test | Success) = 90% Likelihood: P(Positive Test | Failure) = 30% Using Bayes' Theorem: Updated P(Success | Positive Test) = 82% ``` ### 3. Compound Probabilities Multiple events that must occur together: #### Example: Product Launch Success ``` Required for success: - Development completed on time: 80% - Regulatory approval obtained: 70% - Marketing campaign effective: 85% - No major competitor launch: 60% Combined success probability: 0.8 × 0.7 × 0.85 × 0.6 = 28.6% ``` #### Example: Career Advancement ``` Path to promotion: - Complete project successfully: 90% - Manager remains supportive: 85% - No company reorganization: 70% - Budget available for promotion: 80% Combined probability: 0.9 × 0.85 × 0.7 × 0.8 = 42.8% ``` ## 🛠️ Practical Estimation Tools ### Probability Estimation Worksheet ``` PROBABILITY ESTIMATION WORKSHEET Event: _________________________ Date: _________________________ INFORMATION SOURCES: □ Historical data: ________________ □ Reference class: ________________ □ Expert opinion: ________________ □ Personal experience: ____________ BASE RATE ANALYSIS: Similar situations: _______________ Success rate: ___________________ Sample size: ___________________ Relevance to current situation: ___/10 ADJUSTMENTS: Unique factors that increase probability: 1. ____________________________ 2. ____________________________ 3. ____________________________ Unique factors that decrease probability: 1. ____________________________ 2. ____________________________ 3. ____________________________ ESTIMATION METHODS: Method 1 - Direct estimate: ______% Method 2 - Frequency format: ___/___ Method 3 - Three-point: Optimistic: _____% Most likely: _____% Pessimistic: _____% Weighted avg: _____% FINAL ESTIMATE: ______% CONFIDENCE LEVEL: [Low/Med/High] REVIEW DATE: ___________________ ``` ### Quick Reference Probability Ranges | Description | Probability Range | Example Events | |-------------|------------------|----------------| | Almost Certain | 90-99% | Sun rising tomorrow | | Very Likely | 70-89% | Getting to work on time | | Likely | 55-69% | Completing project on schedule | | Possible | 35-54% | Getting promoted this year | | Unlikely | 15-34% | Winning competitive bid | | Very Unlikely | 5-14% | Major system failure | | Almost Impossible | 1-4% | Winning lottery | ### Calibration Testing Kit #### Weekly Calibration Practice ``` Monday: Make 10 predictions about week ahead - Meeting will start on time: ____% - Stock market will go up: ____% - Project deliverable ready: ____% Friday: Score your predictions - Calculate calibration accuracy - Identify bias patterns - Adjust estimation approach ``` #### Domain-Specific Calibration ``` Business Decisions: Track predictions about: - Sales targets hit - Project completion dates - Customer satisfaction scores - Market response to changes Personal Decisions: Track predictions about: - Weather forecasts - Travel delays - Social event attendance - Personal goal achievement ``` ## 🚨 Common Probability Pitfalls ### 1. The Planning Fallacy **Problem:** Underestimating time and overestimating success ``` ❌ Common Error: "90% chance we'll finish on time" Reality: Most projects are delayed ✅ Better Approach: "Historical data shows 60% on-time completion" "Our project has complexity factors → 45% probability" ``` ### 2. Availability Heuristic **Problem:** Recent events feel more likely ``` ❌ After airline crash: "Flying is dangerous, 30% chance of problems" Reality: Flying remains statistically very safe ✅ Corrected View: Use long-term safety statistics, not recent news ``` ### 3. Conjunction Fallacy **Problem:** Thinking specific scenarios are more likely than general ones ``` ❌ Logical Error: "Jane is active in feminist causes and likes reading" P(Jane is bank teller AND feminist) > P(Jane is bank teller) This is mathematically impossible! ✅ Correct Logic: Specific combinations are always less likely than components ``` ### 4. Base Rate Neglect **Problem:** Ignoring underlying frequencies ``` ❌ Ignoring Base Rates: "This startup has great tech, 90% chance of success" Base rate: 90% of startups fail ✅ Incorporating Base Rates: "Good tech improves odds, but base rate is 10% success" "Adjusted estimate: 20-25% chance of success" ``` ## 🎯 Probability in Practice ### Example 1: Investment Decision Tree ``` Investment Opportunity: Tech Startup Base Information: - Industry success rate: 15% - This startup stage: Seed funding - Your investment: $10,000 Probability Assessment: Success Factors: + Experienced team: Increases odds by 5% + Large market: Increases odds by 3% + Unique technology: Increases odds by 2% Risk Factors: - Competitive market: Decreases odds by 3% - Regulatory uncertainty: Decreases odds by 2% Adjusted Probability: Base: 15% Adjustments: +10% - 5% = +5% Final Estimate: 20% chance of success Expected Value: Success (20%): 10x return = $100,000 Failure (80%): Total loss = $0 Expected: (0.2 × $100,000) + (0.8 × $0) = $20,000 Investment worth: $10,000 Expected profit: $10,000 (positive expected value) ``` ### Example 2: Career Decision Tree ``` Decision: Accept promotion to management role Probability Assessment needed for: Management Success: Historical data: 70% of technical people succeed in management Personal factors: + Strong communication skills: +10% + Previous leadership experience: +15% - Prefer technical work: -10% - High stress sensitivity: -5% Adjusted probability: 80% Team Acceptance: Reference class: New managers in similar companies Base rate: 65% get team buy-in within 6 months Personal factors: + Know team well: +20% + Respected by peers: +10% - Lack formal authority experience: -5% Adjusted probability: 90% Career Advancement: Industry data: Management roles lead to faster advancement Success if management works: 85% Recovery if management fails: 60% ``` ## 🔗 Next Steps ### Continue Learning - [[Decision Helper/01-decision-methods/03-decision-tree/examples|See Complete Examples →]] - [[01-decision-methods/04-markov-chain/index|Explore Markov Chains →]] ### Related Topics - [[05-reference/probability-tables|Probability Reference Tables]] - [[04-tutorials/uncertainty-handling|Advanced Uncertainty Techniques]] --- **Next:** [[Decision Helper/01-decision-methods/03-decision-tree/examples|Apply What You've Learned →]]