# Prob and Stats2: Conditional Probability

Conditional Probability & Bayes Theorem

Conditional probability is important because a lot of supervised learning problem is structured in a way like what is Y given features X. And conditional probability is an intuitive way to construct it.

**P(+|A) = 1-P(-|A)**

## Independence

## Conditional Independence

## Bayes Theorem

## Conditional Version of Bayes Theorem

**Questions**

**Monty Hall**

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say №1, and the host, who knows what’s behind the doors, opens another door, say №3, which has a goat. He then says to you, “Do you want to pick door №2?” Is it to your advantage to switch your choice?

## Seattle Rain

You’re headed to Seattle. You want to know if you should bring an umbrella so you call 3 random friends who live there and ask each independently whether it’s raining or not. Each friend has a 2/3 chance of telling you the truth and a 1/3 chance of lying (so mean!). All 3 friends tell you “Yes, it’s raining”. What is the probability that it’s actually raining in Seattle?

## Defective Machine

## Coin in Box

## Coin in Box (2)

**Answers for above questions can be found in Probability and Statistics by Morris H. Degroot. Thanks for reading the post. Hope it is useful!**