Section 4.1: The Nature of Probability
Probability is the chance that something will happen - or we could say probability is how likely
it is that an event will happen.
There are three types of probability.
1) Subjective Probability
2) Empirical Probability
3) Theoretical Probability
I. Subjective Probability. This is probability based on ones’, (possibly educated), guess.
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 an 80% chance that it I will get an A on the test covering chapter 4.
This is nothing more than an educated guess. The estimate of the probability is only as good as
the person making the guess.
II. Empirical Probability. This is probability which is experimentally determined. We use
numbers gathered in some fashion and estimate probabilities based on the information
obtained. If the data being used isn’t obtained in a method that will create quality estimates
our probability estimate may be inaccurate.
Empirical Probability Formula
𝑷(𝑬) =
𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒐𝒃𝒔𝒆𝒓𝒗𝒂𝒕𝒊𝒐𝒏𝒔 𝒊𝒏 𝒘𝒉𝒊𝒄𝒉 "𝑬" 𝒉𝒂𝒔 𝒐𝒄𝒄𝒖𝒓𝒆𝒅
𝒕𝒐𝒕𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒐𝒃𝒔𝒆𝒓𝒗𝒂𝒕𝒊𝒐𝒏𝒔
Where:
P(E) = probability that an event, E, will occur.
Numerator – number of times the event has occurred
Denominator – total number of events observed
I will work through an example to help you understand the formula. Essentially it tells me to
create a fraction. The numerator has the number of times the desired outcome has happened
and the denominator equals the total number of outcomes.
Example. 100 GCC North students were asked what high school they attended. The results are
given in the table below.
Survey Results
High
School
Sandra Day
26
Mountain Ridge
20
Boulder Creek
14
Sunrise Mountain
10
Other
30
(a) Find the empirical probability that a randomly selected GCC North student went to
Mountain Ridge? (Write your answer as a reduced fraction.)
The formula tells me that my answer is going to be a fraction.
The numerator will be the number of people that went to Mountain Ridge.
The denominator will be the 100 people that were interviewed.
20/100
Answer: 1/5
(b) Find the empirical probability that a randomly selected GCC North student did not go to
Sunrise Mountain? (Write your answer as a reduced fraction.)
The numerator will consist of each student that did not go to Sunrise Mountain. It is easiest to
subtract the number of students that went to Sunrise from 100. 100 – 10 = 90 students didn’t
go to Sunrise.
The denominator will be the 100 students surveyed.
90/100
Answer: 9/10
c) If 500 GCC North students were asked which high school they attended how many would say
Sandra Day?
I need to multiply the probability a student went to Sandra Day times the 500 students that are
being asked in this part of the question to get my answer.
26
100
∗ 500 = 130
Answer: 130 students would say they attended Sandra Day.
Homework #1-8: Write your answer as a reduced fraction.
1) In a given week, a veterinarian treated the following animals:
Animal
Number treated
Dog
40
Cat
35
Bird
15
Iguana
5
Total
95
Determine the empirical probability that the next animal she treats is
a) A dog
b) A cat
c) An iguana
2) Mr. Schwarz’s grade distribution over the past 3 years for a course in College Algebra is
shown below.
Grade
Number
A
43
B
182
C
260
D
90
F
62
Total
637
If Gilligan plans to take College Algebra with Mr. Schwarz, determine the empirical probability
he receives a grade of
a) A
b) C
c) D or higher
3) A survey was given to 250 people at a Sun’s game. 30 of them said Charles Barkley was their
favorite Sun of all time.
a) If another person at the game is asked to name their favorite Sun of all time what is the
probability they will select Charles Barkley?
b) Estimate the number of people of the 12,000 in attendance will say Charles Barkley is their
favorite Sun’s player of all time?
4) A manufacturer tested 1000 radios at random and found 15 of them to be defective.
a) Determine the empirical probability a radio is defective.
b) Estimate the number of defective radios in a shipment of 5,000 radios.
5) John has taught 500 MAT 142 students. 220 of his MAT 142 students received a grade of an
A.
a) Determine the empirical probability that a student in John’s MAT 142 class receives a grade
of an A.
b) Estimate the number of A’s John will give in a MAT 142 class that has 30 students. (Round to
the nearest whole number)
6) Charles has attempted 80 free throws this season. Charles has made 50 and missed 30.
a) What is the empirical probability that Charles will make his next free throw?
b) Estimate the number of free throws Charles will make during a game that he attempts 20
free throws. (Round to the nearest whole number)
7) A survey was conducted to determine students’ favorite breeds of dogs. Each student chose
one breed.
Dog
Collie
Spaniel
Lab
Boxer
PitBull
Other
#
10
15
35
8
5
12
a) What is the empirical probability that a students’ favorite dog is a boxer? (Hint 85 students’
were surveyed)
b) If 200 students are asked to name their favorite dog. Estimate how many of them will
choose a pit bull.
8) A survey was conducted to determine students’ majors. Each student chose one major.
major
Nursing
Business
Education
Psychology
Math
Other
#
50
30
10
5
1
4
a) What is the empirical probability that a students’ major is education?
b) If 40 students are asked to name their major. Estimate how many of them will choose a
Business. (Round to the nearest whole number)
III. Theoretical Probability. This is probability which is based on mathematical theories and
formulas.
We can answer theoretical probability questions without knowing a lot of theory, although it
will be nice to know some theory and notation. In fact solving theoretical probability questions
is basically identical to solving empirical probability questions.
Here is an example of a theoretical probability question that we can attempt to solve without
knowing any theory.
Example of a theoretical probability question: 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.
a) Find the theoretical probability that the marble is red. (Write your answer as a reduced
fraction.
I will still use the same formula. The numerator will be the number of ways the desired
event can occur. The numerator will be 50 as there are 50 red marbles.
The denominator needs to be the total number of possible outcomes when I pull a marble
from the jar. The denominator will be 200 as the jar contains 200 marbles.
Answer: 50/200 = ¼
b) Find the theoretical probability that the marble is red or green: (Write your answer as a
reduced fraction.
The numerator will be the number of ways the desired event can occur. The numerator
will be 70 which counts the 50 red marbles and 20 green marbles.
The denominator needs to be the total number of possible outcomes when I pull a
marble from the jar. The denominator will be 200 as the jar contains 200 marbles.
Answer: 70/200 = 7/20
c) Find the theoretical probability that the marble is red and green: (Write your answer as a
reduced fraction.
The numerator will be the number of ways the desired event can occur. The numerator will
be 0 as there is no marble that is both red and green.
The denominator needs to be the total number of possible outcomes when I pull a marble
from the jar. The denominator will be 200 as the jar contains 200 marbles.
Answer: 0/200 = 0
If this problem was an empirical probability problem we wouldn’t know the exact contents of
the jar and would have to do an experiment to help predict the contents of the jar. The
empirical probability version of the problem may read like this.
Example: A marble is removed from a jar and its color is noted. The marble is placed back in
the jar and another one is selected. This process is repeated 30 times. The result of the
experiment is 15 blue, 8 red, 5 yellow and 2 green.
What is the probability the next marble drawn is red?
We would form a fraction to answer this question.
The numerator would be the 8 red marbles that were drawn during the experiment.
The denominator would be the 30 marbles that were drawn during the experiment.
Answer: 8/30 = 4/15
There are a handful of terms that we should be familiar with for the next group of problems.
You certainly can solve the problems without knowing each definition. I almost feel like this
page of definitions is a bit much at this point in our discussion about probability. You should
make an effort to understand the terms, but I won’t ask definitions on the test.
 A probability experiment is a controlled operation that yields a set of measurable
results.
Example of a “probability experiment”:
A dice is rolled one time and the number that is on the top of the dice is recorded.
(This is a probability experiment as it can be performed in a controlled fashion and the
results are measurable.)
 Outcomes: 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.
In our example each time the dice is rolled and a number is recorded is called a trial.
 The set of all possible outcomes of an experiment is called the sample space. The
sample space is usually denoted by the capital letter S and it is often written in set
roster form.
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}
 An event is a subset of the sample space. There are two types of events: simple events
and compound events.
 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 = {1} (rolling a 1)
E2 = {2} (rolling a 2)
E3 = {3}
E4 = {4}
E5 = {5}
E6 = {6}
 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 = event a number greater than 7 is rolled.
I={ }
 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 theoretical probability formula applies:
𝑛(𝐸)
P(E) = 𝑛(𝑆) or in words
𝑃(𝐸𝑣𝑒𝑛𝑡) =
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑤𝑎𝑦𝑠 𝑡ℎ𝑒 𝑒𝑣𝑒𝑛𝑡 𝑐𝑎𝑛 𝑜𝑐𝑐𝑢𝑟
𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑖𝑚𝑝𝑙𝑒 𝑒𝑣𝑒𝑛𝑡𝑠
Example: A 6 sided dice is rolled and the number that is face up is recorded.
a) Let O be the compound event that the number on the dice is odd. Find P(O)
I need to form a fraction.
The numerator will be the number of elements in set “O” that is the number of odd numbers
that can be rolled. The numerator will be 3, as there are 3 odd numbers that can be rolled
(1,3,5).
The denominator will be the 6 numbers that can come up when I roll a dice.
Answer: 3/6 = 1/2
b) Let I be the event that the number on the dice is greater than 7. Find P(I)
The numerator will be the number of ways a number greater than 7 can occur. The numerator
will be 0 as it is impossible to roll a number greater than 7.
The denominator will be the 6 numbers that can come up when I roll a dice.
Answer: 0/6 = 0
(This problem demonstrates the theory that the lowest a probability can be is 0. Probabilities
can’t be negative numbers.)
c) Let W be the event a number less than 7 is rolled. Find P(W)
The numerator will be the number of ways a number less than 7 can occur. The numerator will
be 6 as any of the 6 numbers that can be rolled are less than 7.
The denominator will be the 6 numbers that can come up when I roll a dice.
Answer: 6/6 = 1
(This problem demonstrates the theory that the highest a probability can be is 1 or 100%.)
Homework #9 – 22: Write your answer as a reduced fraction.
9) 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 odd number?
c)
What is the probability you press a key for a number less than 7?
10) A keypad contains the Roman Numerals: I V X L C D M. If one key is selected at random,
a) What is the probability the “V” is selected?
b) What is the probability the number is greater than 60?
c) What is the probability that the number is evenly divisible by 10?
Problems #11 - 30: one card is selected from a deck of cards. Find the probability the card
selected is: Write your answer as a reduced fraction. (a diagram of a deck of cars is located at
the end of this section)
11) a seven
12) a three
13) a heart
14) a spade
15) a black card
16) a red card
17) a seven and a red card
18) a queen and a black card
19) a red card and a black card
20) a spade and a red card
21) a black card and a seven
22) a spade and a seven
23) a diamond and a four
24) a red card and a four
25) a queen or a king
26) a two or a jack
27) a queen and a king
28) a queen and a jack
29) a club or a spade
30) a jack or a king
Homework #31 – 34: A dart is thrown at this target and the color it lands on is noted. Find the
requested probabilities. Write your answer as a reduced fraction.
31) Probability that the dart lands in the blue region.
32) Probability the dart lands in red region
33) Probability the dart does not land in blue region.
34) Probability the dart does not land in red region.
Homework #35-38: The following spinner is spun one time. It is spun again if it lands on a line.
Find the requested probabilities. Write your answer as a reduced fraction.
35) Probability that spinner lands in the blue region (this blue looks fairly purple to me).
36) Probability the spinner lands in red region
37) Probability the spinner does not land in blue region.
38) Probability the spinner does not land in red region.
#39 – 42: 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: Write your answer as a reduced fraction.
39) The letter s is not selected
40) The letter i or p is selected
41) The letter w is selected
42) The letter w is not selected
#43 – 46: 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
American
260
273
533
Foreign
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: Write your answer as a reduced fraction.
43) A man.
44) Drives an American car.
45) A woman that drives a foreign car.
46) A man that drives an American car.
#47-50: refer to the table which shows the results of a survey regarding the type of cell phone
owned by people who were interviewed. Write your answer as a reduced fraction.
men
Women
total
IPhone
150
200
350
Galaxy
100
50
150
Other
125
175
300
Total
375
425
800
If one person who completed the survey is selected at random, determine the probability the
person selected is
47) A man
48) Owns an IPhone
49) Is a woman that doesn’t own an IPhone
50) Does not own a Galaxy
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)
Answers to odd numbered problems.
1a) 4/19
3a) 3/25
5a) 11/25
7a) 8/85
9a) 1/10
11) 1/13
19) 0
27) 0
35) 2/3
43) 69/143
1b) 7/38
1c) 1/38
3b) 1440 people
5b) 13
7b) 12
9b) ½
9c) 7/10
13) ¼
15) ½
21) 1/26
23) 1/52
29) 1/2
31) ¼
37) 1/3
39) 7/11
45) 97/715
47) 15/32
17) 1/26
25) 2/13
33) ¾
41) 0
49) 9/32