(4.1, 4.2, 4.3) Set Theory, Intro to Probability
SET: collection of elements S={…}
SUBSET: a set in which every element is also contained in a larger set.
ELEMENT: an single item, it may be in a set or not in a set, usually denoted ω
S={a, b, c, d, e}
S is a SET containing 5 elements
N={a, b, c}
a is an ELEMENT of S, a  S
d is NOT an ELEMENT of N, d  N
N is a SUBSET of S, N  S
RANDOM EXPERIMENT: an action whose outcome cannot be predicted with
CERTAINTY beforehand
SAMPLE SPACE (Ω): the set of ALL POSSIBLE outcomes for a random
experiment, with Ω representing the entire sample space and ω representing
each element in that sample space.
Sample space and population are similar in that they contain “all possibilities”
– the key difference is that a sample space is unique to a random experiment
and a population is unique to sampling (usually people).
EVENT: a collection of sample points. It OCCURS if the outcome is an
element, ω, of the sample space Ω
Example:
Random Experiment: roll a fair 6-sided dice
Sample Space: {1, 2, 3, 4, 5, 6}
Event: Roll at least 4, the subset (outcomes for this event) would be: {4, 5, 6}
If all outcomes are equally likely:
P(Event) = N(Event)/N(Total) = 3/6 = 1/2
Where “N” denotes the “number” of elements in the event and the number of
elements total (or in the sample space).
You toss a fair coin 3 times:
1. What is the sample space?
2. Find the following probabilities:
a. P(exactly one Head)
b. P(at least one Head)
c. P(exactly one Head or exactly one tail)
d. P(no heads)
e. P(no heads or no tails)
f. P(at least one head AND at least one tail)
g. P(3 heads|1st toss is a head)
h. P(at least 2 heads|1st toss is a head)
i. P(at least 1 tail|1st toss is a tail)
j. P(all 3 tails|1st toss is a head)
VENN DIAGRAMS
Ω = the UNIVERSAL set, represented by a rectangle. This set includes
EVERYTHING! All other sets are included in this set.
A set inside the universal set is denoted by a circle (labeled with the set name)
Mutually Exclusive
Exhaustive Events
No Elements in common
Events partition the
outcome space
Plain M&M’s and Peanut
M&M’s
Male/Female
Subsets
Overlap
All elements of
B are also
elements of A
Purdue
Undergrads,
Freshman
Some elements
in common
Freshman,
Management students
Complement
Union
Intersection
All elements NOT in A,
denoted AC
All elements in either A
or B, denoted AUB
All elements in both A
and B, denoted A∩B
Example:
In a certain population,
10% of the people are rich
8% are famous and
3% are rich and famous
Draw a Venn Diagram of the situation and label ALL probabilities:
**Don’t forget the rectangle representing the population and the probability of those that are
not rich nor famous**
What is the probability a randomly chosen person is not rich?
P(RC) = 100% - 10% = 90%
What is the probability a randomly chosen person is rich but not famous?
P(R and FC) = P(R) – P(R AND F) = 10% - 3% = 7%
What is the probability a randomly chosen person is either rich or famous?
P(R U F) = P(R) + P(F) – P(R and F) = 10% + 8% - 3% = 15%
What is the probability a randomly chosen person is rich or famous but not
both?
P((R and FC) U (F and RC)) = 7% + 5% = 13%
What is the probability they have at least wealth or fame?
P(R U F) = 15%
What is the probability they have neither wealth nor fame?
P(R U F)C = 1 – P(R U F) = 100% - 15% = 85%
Complement Rule: for each event E, P(E) = 1 – P(EC)
Addition Rule: for any two events A and B, P(AUB) = P(A) + P(B) – P(A∩B)
Why do we need to subtract? When does it REALLY matter?
OVERLAP!! We end up counting the overlap TWICE when we add the
entire A circle and B circle, so we need to subtract the overlap so its
only counted ONCE.
DeMorgan’s Laws
(AUB)C = AC ∩ BC
(A∩B)C = AC U BC
Inclusion – Exclusion Principle:
P(AUBUC) = P(A) + P(B) + P(C) – P(A∩B) – P(A∩C) – P(B∩C) + P(A∩B∩C)
Example:
You are throwing a party with 75 guests.
43 enjoy an Active game (A - such as catch/frisbee/touch football),
37 enjoy board games (B),
36 enjoy cards (C),
10 enjoy an Active game AND Cards, but NOT board games.
11 will ONLY play cards,
8 will play all three games and
5 will not play ANY of the games.
Create a Venn Diagram to summarize the data
How many people like to play at least one game?
75 – 5 = 70
How many people like to play exactly 2 games?
10 + 13 + 7 = 20
You roll two fair 4-sided dice.
1. What is the sample space?
2. Find the following Probabilities:
a. P(sum is 8)
b. P(sum is at least 6)
c. P(sum is 4 or 7)
d. P(first die is a 1 and second die is a 2)
e. P(first die is a 1 and sum is 3)
f. P(at least one of the dice is a 4)
g. P(sum 5|first die is 4)
h. P(sum at least 3|first die is a 1)
**Hand out Transparencies**
Let A be an event that happens 40% of the time
Let B be an event that happens 75% of the time
1. What is the SMALLEST probability A  B can have?
2. What is the LARGEST probability A  B can have?
3. What is the SMALLEST probability A  B can have?
4. What is the LARGEST probability A  B can have?