Download Intro to Probability

Introduction to Probability Day 11
Three Types of Probability
I. Subjective Probability. This is probability based on ones’,
(possibly educated), beliefs.
Example. Your sister says that there is a 90% chance that she will be the
lead in the school play next year.
Example. The weather forecast is 20% chance of rain.
II. Empirical Probability. This is probability which is
experimentally determined.
Example. The following survey was given to 100 GCC sophomores:
1. (Yes or No.) Do you consider your experience in math classes
at GCC an overall positive experience?
2. (Yes or No.) Have you taken more than one math
class at GCC?
Survey Results
Positive
Experience
Negative
Experience
Taken more than one
26
15
Taken no more than one
20
39
Find the empirical probability that a GCC math student has (a) taken
more than one math class at GCC; (b) had a negative experience
in math classes at GCC; (c) has taken more than one math class or
had a positive experience; (d) has taken more than one math class
and had a negative experience.
Three Types of Probability
III. Theoretical Probability. This is probability which is based on
mathematical theories and formulas.
Example. A jar contains 80 white marbles,
100 blue marbles, 92 red marbles, 70 yellow
marbles, and 85 green marbles. One marble
is drawn from the jar at random. Find the
theoretical probability that the marble is
(a) red.
(b) red or green.
(c) red and green.
(d) neither red nor green.
Definitions. (Single die example done in class with this.)
 A probability experiment is a controlled operation that yields
a set of results.
 The results of a probability experiment are called outcomes.
 Each time an experiment is performed it is called a trial.
 The set of possible outcomes of an experiment is called the
sample space. The sample space is usually denoted by the
capital letter S.
 An event is a subset of the sample space.
 An event containing exactly one element is called a simple event.
 An event containing more than one element is called a compound
event.
 The event of the empty set is called an impossible event.
 The event of the sample space S is called a sure event.
 If all simple events in a probability experiment are equally likely, and E
is a given event, then the following probability formula applies:
P(E) = n(E)/n(S)
Properties of Probability
1. For all events E ⊆ S,
0  P( E )  1 ,
where P(φ) = 0 and P(S) = 1.
2. If E1 , E2 ,..., En is a list of all simple events in a sample space
S, then
P( E1 )  P( E2 )  P( E3 )   P( En )  1
Example. A fair coin is tossed two times.
(a) Give the sample space S in roster form.
S = {HH, HT, TH, TT}
(b) Let E be the event of two heads. List E in
roster form.
E={HH}
(c) Let F be the event of at least one head. List F in roster form.
F = {HH, HT, TH}
(d) Find P(E) and P(F), (the probabilities of E and F).
P(E) = n(E)/n(S) = ¼ = .25 = 25%
P(F) = n(F)/n(S) = ¾ = .75 = 75%
Example: The cards below are shuffled and one card is drawn at
random.
(a) List the sample space and the event of a red card in roster form.
(b) List the event of an even card in roster form.
Face Cards
Red Cards
Heart Diamond
Black Cards
Club Spade
Ace
(A,H)
(A,D)
(A,C)
(A,S)
King
(K,H)
(K,D)
(K,C)
(K,S)
Queen
(Q,H)
(Q,D)
(Q,C)
(Q,S)
Jack
(J,H)
(J,D)
(J,C)
(J,S)
10
(10,H)
(10,D)
(10,C)
(10,S)
9
(9,H)
(9,D)
(9,C)
(9,S)
8
(8,H)
(8,D)
(8,C)
(8,S)
7
(7,H)
(7,D)
(7,C)
(7,S)
6
(6,H)
(6,D)
(6,C)
(6,S)
5
(5,H)
(5,D)
(5,C)
(5,S)
4
(4,H)
(4,D)
(4,C)
(4,S)
3
(3,H)
(3,D)
(3,C)
(3,S)
2
(2,H)
(2,D)
(2,C)
(2,S)
1
2
3
4
5
6
1
(1,1)
(1,2)
(1,3)
(1,4)
(1,5)
(1,6)
2
(2,1)
(2,2)
(2,3)
(2,4)
(2,5)
(2,6)
3
(3,1)
(3,2)
(3,3)
(3,4)
(3,5)
(3,6)
4
(4,1)
(4,2)
(4,3)
(4,4)
(4,5)
(4,6)
5
(5,1)
(5,2)
(5,3)
(5,4)
(5,5)
(5,6)
6
(6,1)
(6,2)
(6,3)
(6,4)
(6,5)
(6,6)
Law of Large Numbers
The Law of Large Numbers is a proven
probability theory that states that if an
experiment is performed many many times,
then the empirical probabilities will be
super close to the theoretical probabilities
on average.
More Problems
 Monty Hall Problem
 http://www.cut-the-knot.org/hall.shtml