Conditional Probability and Bayes' Theorem: A Comprehensive Guide with Examples, Lecture notes of Probability and Statistics

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Conditional Probability

Mutually Exclusive Events

P (E [ F ) = P (E) + P (F )

If events are mutually exclusive, probability of OR is simple:

Today’s Lesson

• Roll two 6-sided dice, yielding values D

1

and D

2

• Let E be event: D

1

+ D

2

• What is P(E)?

§ |S| = 36, E = {(1, 3), (2, 2), (3, 1)}

§ P(E) = 3/36 = 1/

• Let F be event: D

1

• P(E, given F already observed)?

§ S = {(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)}

§ E = {(2, 2)}

§ P(E, given F already observed) = 1/

Dice – Our Misunderstood Friends

• Conditional probability is probability that E occurs

given that F has already occurred “Conditioning on F”

• Written as

§ Means “P(E, given F already observed)”

§ Sample space, S, reduced to those elements

consistent with F (i.e. S Ç F)

§ Event space, E, reduced to those elements

consistent with F (i.e. E Ç F)

Conditional Probability

P (E|F )

With equally likely outcomes:

P(E | F) =

of outcomes in E consistent with F

of outcomes in S consistent with F

F

EF

SF

EF

Conditional Probability

P (E|F ) =

P (E) =

• General definition of Chain Rule:

1 2 3 n

P E E E E

1 2 1 3 1 2 1 2 - 1

n n

P E P E E P E E E P E E E E

Generalized Chain Rule

• Learn

What is the probability that a user will watch Life is Beautiful? P ( E )

Netflix and Learn

What is the probability that a user will watch Life is Beautiful? P ( E )

Netflix and Learn

P ( E ) = 10,234,231 / 50,923,123 = 0.

P (E) = lim n! n(E) n

#people who watched movie #people on Netflix

Netflix and Learn

What is the probability that a user will watch Life is Beautiful, given they watched Amelie? P ( E|F ) P (E|F ) =

P (EF )

P (F )

#people who watched both #people on Netflix #people who watched F #people on Netflix

P (E|F ) =

P (EF )

P (F )

#people who watched both #people on Netflix #people who watched F #people on Netflix

Netflix and Learn

P ( E|F )

What is the probability that a user will watch Life is Beautiful, given they watched Amelie? P (E|F ) =

P (EF )

P (F )

#people who watched both #people on Netflix #people who watched F #people on Netflix

Netflix and Learn

Let E be the event that a user watched the given movie, Let F be the event that the same user watched Amelie: P ( E | F ) =

P ( E | F ) =

P ( E | F ) =

P ( E | F ) =

P ( E | F ) =

Machine Learning

Machine Learning is:

Probability + Data + Computers