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!