Section 12.1 and 12.2 class notes
Section 12.1: The Nature of Probability
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. I think there is a 80% chance that it I will get a A on the test covering chapter 12.
II. Empirical Probability. This is probability which is experimentally determined. We use numbers
gathered and estimate probabilities based on the information obtained. The probability estimate may
not be that good if the data being used isn’t obtained in a method that will create quality estimates. For
example if this survey was given to students in a Calculus class the data might not reflect reality.
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;
Here are a few problems from section 12.1
17. In a given week, a veterinarian treated the following animals:
Animal
Dog
Cat
Bird
iguana
Number treated
40
35
15
5
Determine the probability that the next animal she treats is
a)
b)
c)
A dog
A cat
An iguana
22. Mr. Doole’s grade distribution over the past 3 years for a course in college algebra is shown below.
grade
A
B
C
D
F
I
Number
43
182
260
90
62
8
If Sue Gilligan plans to take college Algebra with Mr. Doole, determine the empirical probability she
receives a grade of
a)
b)
c)
A
C
D or higher
27. An experimental serum was injected into 500 guinea pigs. Initially, 150 of the guinea pigs had
circular cells, 250 had elliptical cells, and 100 had irregularly shaped cells. After the serum was injected
none of the guinea pigs with circular cells was affected, 50 with elliptical cells were affected, and all of
those with irregular cells were affected. Find the empirical probability that a guinea pig with (a) circular
cells, (b) elliptical cells, and (c) irregular cells will be affected by injection of the serum.
III. Theoretical Probability. (start of section 12.2) This is probability which is based on mathematical
theories and formulas.
Example. A jar contains 40 white marbles, 60 blue marbles, 50 red marbles, 30 yellow marbles, and 20
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

A probability experiment is a controlled operation that yields a set of results.
Let me define an experiment as rolling a dice one tie and writing down the number that is on the top
of the dice.

The results of a probability experiment are called outcomes.
Possible outcomes for this experiment are rolling a 1 or 2 or 3 or 4 or 5 or 6.

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.
I can roll any of the numbers 1,2,3,4,5 or 6 so we write the sample space S as follows
S = { 1,2,3,4,5,6} (notice we use set notation and call the set S meaning sample space)

An event is a subset of the sample space.

An event containing exactly one element is called a simple event.
These are the six simple events for this “experiment”. I listed and named the events using a
standard naming strategy.
E1 = rolling a 1
E2 = rolling a 2
E3 = rolling a 3
E4 = rolling a 4
E5 = rolling a 5
E6 = rolling a six

An event containing more than one element is called a compound event.
This is in an example of a compound event
O = event the an odd number is rolled
O can be written as the following set O = {1,3,5}

The event of the empty set is called an impossible event.
This is an example of an impossible event
I={ }
I = event a number greater than 7 is rolled.

The event of the sample space S is called a sure event.
Here is an example of a sure event. W = event a number less than 7 is rolled.
In set notation W = {1,2,3,4,5,6}

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) or in words
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑡ℎ𝑒 𝑒𝑣𝑒𝑛𝑡 𝑐𝑎𝑛 𝑜𝑐𝑐𝑢𝑟
𝑃(𝐸𝑣𝑒𝑛𝑡) = 𝑡ℎ𝑒 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑖𝑚𝑝𝑙𝑒 𝑒𝑣𝑒𝑛𝑡𝑠 𝑡ℎ𝑎𝑡
For the events discussed
3
6
P(O) = 𝑜𝑟
1
2
0
P(I) = 6 = 0
6
P(W) = 6 = 1
These Properties of Probability follows from the previous discussion. (This is rule is written using proper
mathematics. I tried to explain the meaning behind this rule after the symbols.
For all events E ⊆ S,
0  P( E )  1
where P(φ) = 0 and P(S) = 1.
This says that any event can’t have a probability less than 0 or greater than 1. It tells me the probability
of any event is between 0 and 1 inclusive. This tells me that the probability of something that is
impossible is 0, and the probability of something that is certain to happen is 1 or 100%.
For example: The probability of rolling a six sided dice and getting a 7 is 0
The probability of rolling a six sided dice and getting a number less than 7 is 1 or 100%.
.
Here are some problems from the text that use the properties of probability.
Section 12.2 Problem 14
A TV remote control has keys for channels 0 through 9. If you select a key at random,
a)
What is the probability you press channel 6?
b)
What is the probability you press a key for an even number?
c)
What is the probability you press a key for a number less than 7?
Section 12.2 problems 17 – 26, one card is selected from a deck of cards. Find the probability the card
selected is: (FOR A LISTING OF CARDS IN A STANDARD DECK OF 52 CARDS GO AHEAD 2 PAGES)
18) a 7 or a 9
20) the five of diamonds
22) a heart
24) a red card and a black card
26) a jack and a heart
Let’s try problems 28 and 30 from the text. I can’t duplicate these very well.
Section 12.2 problems 43 – 46, a traffic light is red for 25 seconds, yellow for 5 seconds and green for 55
seconds. What is the probability that when you reach the light,
44) the light is yellow
46) the light is not green
Section 12.2 #47 – 52, each individual letter of the word Mississippi is placed on a piece of paper, and all
11 pieces of paper are placed in a hat. If one letter is selected at random from the hat, find the
probability that
48) the letter s is not selected
50) the letter i or p is selected
52) the letter w is selected
Section 12.2 # 61 – 66, refer to the table which shows the results of a survey regarding the manufacturer
of cars driven by the people who were interviewed.
men
Women
total
GM, Ford, Chrysler
260
273
533
Other
85
97
182
Total
345
370
715
If one person who completed the survey is selected at random, determine the probability the person
selected is
62) a man
64) drives a car manufactured by a company other than GM, Ford or Chrysler
This shows the cards in a standard deck of 52 cards. J Q K are face cards, ace is not a face card.
Red Cards
Black Cards
Heart
Diamond
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)
This is the sample space for rolling 2 dice. The sample space shows the number on dice one followed by
the number on dice 2.
DICE 1
→
1
2
3
4
5
6
Dice 2
↓
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)