Rules of Probability
Addition Rule: If E1 , E2 , . . . En are mutually exclusive events
(meaning Ei Ej = for all i = j), let
E = E1 E2 . . . En
be the event that at least one of the Ei occurs. Then
P(E ) =
Rules of Probability
Addition Rule: If E1 , E2 ,

Friendship Paradox
Take a group of people. For each person, consider the
numbers:
number of friends the person has,
average number of friends that friends of the person has.
Friendship Paradox
Take a group of people. For each person, consider the
numbers:

Conditional Probability
P(A|B) means the probability of A, given B
Conditional Probability
P(A|B) means the probability of A, given B
P(A|B), P(B|A) and P(A B) all measure the same events.
Conditional Probability
P(A|B) means the probability of A, given B

Certainty Equivalent
A choice is risk averse if the outcome is more certain, but
has a lower () expected value.
Certainty Equivalent
A choice is risk averse if the outcome is more certain, but
has a lower () expected value.
A game: You get to open one of

Announcements
Oce hours are now 2-4 on Tuesdays.
Question 4: What is the probability that AT LEAST one die
lands on 1?
Baseball
Suppose that a baseball player has a batting average of 0.333.
Baseball
Suppose that a baseball player has a batting average of

Playing Cards
See the Handout #3.
From Last Time:
We also had three axioms of probabilities:
From Last Time:
We also had three axioms of probabilities:
Axiom 1: The probability of any event is a nonnegative
number.
Axiom 2: The certain event S has probabi

Administration
email address: [email protected]
course webpage: math.upenn.edu/chhays/math180.html
From Last Time:
Game Show: You choose to either roll or not roll a six-sided
die. Ill pay you $10, 000 times the value of the die OR
$30, 000 if you do

Administration
Homework #1 will be posted online today.
Decision Trees
Decision trees provide a way of organizing decisions
Decision Trees
Decision trees provide a way of organizing decisions
corresponds to choices.
corresponds to chance.
corresponds to