Conditional Probability
Conditioning – the soul of statistics
Independence means multiplication
Newton–Pepys problem: another gambling problem 🎲
Which of the following three propositions has the greatest chance of success?
- A. Six fair dice are tossed independently and at least one “6” appears.
- B. Twelve fair dice are tossed independently and at least two “6”s appear.
- C. Eighteen fair dice are tossed independently and at least three “6”s appear.
Conditional probability 🏄♂️
- How do you update probabilities/beliefs/uncertainty based on new evidence?
- Do you update in a coherent, consistent, and logical manner?
- $P_{(A | B)} = \frac{P_{(A \bigcap B)}}{P_{(B)}}$ if $P_{(B)} > 0$
- $P_{(A \bigcap B)} = P_{(B)} P_{(A|B)} = P_{(A)} P_{(B|A)}$
- $P_{(A | B)} = \frac{P_{(B | A)}P_{(A)}}{P_{(B)}}$ ❤️