Conditional Probability Calculator

Conditional probability represents how our belief in one event changes when we learn another event has occurred. It's the mathematical formalization of 'given this new information, what's the probability?' **Formula**: P(A|B) = P(A ∩ B) / P(B) Where: - P(A|B) = 'probability of A given B' (the conditional probability we want) - P(A ∩ B) = 'probability of A AND B' (joint probability — both events happen) - P(B) = probability of event B occurring **Important**: P(B) must be > 0. You can't condition on an impossible event. **Why This Formula Makes Sense**: When we condition on B, we restrict our 'sample space' from everything possible to just the outcomes where B is true. Within this reduced space, we ask: what fraction also includes A? That fraction is P(A ∩ B) / P(B). **Bayes' Theorem Connection**: Conditional probability leads directly to Bayes' theorem: P(A|B) = P(B|A) × P(A) / P(B) This allows 'reversing' conditional probabilities — going from P(symptoms|disease) to P(disease|symptoms), which is what doctors need to know. **Common Misinterpretation**: P(A|B) ≠ P(B|A)! Example: P(being wet | raining) ≈ 1 (if it's raining, you're almost certainly wet outside) But P(raining | being wet) is much lower (you could be wet from a shower, swimming, etc.) This confusion is called the 'prosecutor's fallacy' when misused in legal contexts. **Independence**: Events A and B are independent if P(A|B) = P(A). In other words, knowing B happened tells you nothing about whether A happened. For independent events: P(A ∩ B) = P(A) × P(B). **Applications**: 1. **Medical Testing**: Given a positive test result, what's the probability the patient actually has the disease? This requires considering both test sensitivity (P(positive|disease)) and disease prevalence (P(disease)). 2. **Weather Forecasting**: Given cloudy skies this morning, what's the probability of rain this afternoon? Uses historical data. 3. **Spam Filtering**: Given certain words appear in an email, what's the probability it's spam? Naive Bayes classifiers use this. 4. **Risk Assessment**: Given market conditions X, what's the probability of investment loss? 5. **Card Games**: Given the first card drawn was a king, what's the probability the next card is also a king? **Famous Example — The Monty Hall Problem**: Three doors, one has a prize. You pick door 1. Monty (who knows) opens door 3, showing it's empty. Should you switch to door 2? Counterintuitively, yes! Your original choice had P(win) = 1/3. The remaining unopened door now has P(win) = 2/3 because Monty's action gave you information. This is a classic conditional probability problem. **Famous Example — The False Positive Paradox**: A disease affects 1 in 1000 people. A test is 99% accurate (99% sensitivity, 99% specificity). You test positive. What's the probability you have the disease? Most people guess 99% — wrong. The actual answer is about 9%. This is because of base rates: out of 1000 people, only 1 has the disease (correctly detected), but ~10 healthy people falsely test positive (1000 × 0.01). So P(disease|positive) = 1/(1+10) = ~9%. This explains why rare disease screening is complex.