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

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Piech, CS106A, Stanford University

Conditional Probability

Recall: S = all possible outcomes. E = the event.

Axioms of Probability

• Axiom 1: 0 £ P ( E ) £ 1
• Axiom 2: P ( S ) = 1
• Axiom 3: P ( E c ) = 1 – P(E) Aside: axiom 3 is often stated as the probability of mutually exclusive events. We’ll come back to that later in the lecture…

Mutually Exclusive Events

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

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

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

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

P (E [ F ) =

Mutually Exclusive Events

If events are mutually exclusive probability of OR is easy!

P ( E

c

) = 1 – P ( E )?

P ( E ∪ E

c

) = P ( E ) + P ( E

c

Since E and E c are mutually exclusive

P (S) = P ( E ) + P ( E

c

Since everything must either be in E or E c

1 = P ( E ) + P ( E

c

P ( E

c

) = 1 – P ( E )

Axiom 2 Rearrange

Piech, CS106A, Stanford University

• 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

Piech, CS106A, Stanford University

Dice – Our Misunderstood Friends

• Two people each roll a die, yielding D 1

and D

2

You win if D

1

+ D

2

• Q: What do you think is the best outcome for D 1

• Your Choices: § A. 1 and 3 tie for best § B. 1, 2 and 3 tie for best § C. 2 is the best § D. Other/none/more than one

Piech, CS106A, Stanford University

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) =

Piech, CS106A, Stanford University

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) =

Shorthand notation for set intersection (aka set “and”)

Piech, CS106A, Stanford University

• 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

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Conditional Paradigm

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Netflix and Learn

P ( E )

S = {Watch, Not Watch} E = {Watch} P(E) = ½? What is the probability that a user will watch Life is Beautiful?

Piech, CS106A, Stanford University What is the probability that a user will watch Life is Beautiful? P ( E )

Netflix and Learn