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.
Slide 3
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?
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. Survey Results Positive
Experience Negative Experience Taken more than one2615 Taken no
more than one2039
Slide 4
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.
Slide 5
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.
Slide 6
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)
Slide 7
Properties of Probability 1. For all events E S,, where P( ) =
0 and P(S) = 1. 2. If is a list of all simple events in a sample
space S, then
Slide 8
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) = =.25 = 25% P(F) = n(F)/n(S) = =.75 = 75%
Slide 9
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.
Slide 10
Red CardsBlack Cards HeartDiamondClubSpade Ace
(A,H)(A,D)(A,C)(A,S) Face Cards 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)
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.
Slide 13
More Problems Monty Hall Problem
http://www.cut-the-knot.org/hall.shtml