Conditional Probability
Updating beliefs with new evidence
🎯 Definition
The probability of event \(E\) occurring, given that event \(F\) has already occurred.
\( P(E|F) = \frac{P(E \cap F)}{P(F)} \)
"The part of E that is inside F, divided by the total size of F"
🔄 Bayes' Theorem
Bayes' Theorem allows us to flip conditional probabilities. It's crucial for medical diagnosis, spam filtering, and AI.
\( P(A|B) = \frac{P(B|A)P(A)}{P(B)} \)
Interactive: Medical Test Paradox
A test is 99% accurate. You test positive. What is the chance you actually have the disease? (Hint: It depends on how rare the disease is!)
Probability you have disease given Positive Test:
Even with high accuracy, if the disease is rare, false positives can outnumber true positives!
🚪 The Monty Hall Problem
Behind one door is a Car 🚗. Behind the others, Goats 🐐. You pick a door. Monty opens another door with a Goat. Should you switch?
Pick a door!
Stay Wins
0
Switch Wins
0
📝 Test Your Understanding
Question 1:
If \(A\) and \(B\) are independent, what is \(P(A|B)\)?