Chapter 4:
Probability: The Study of Randomness
Dr. Nahid Sultana
Chapter 4
Probability: The Study of Randomness
4.1 Randomness
4.2 Probability Models
4.3 Random Variables
4.4 Means and Variances of Random Variables
4.5 General Probability Rules*
4.1 Randomness
 The Language of Probability
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The Language of Probability
Toss a coin, or choose an SRS.
The result can’t be predicted in advance (i.e. it’s uncertain),
because the result will vary when you toss the coin or choose
the sample repeatedly.
But there is however a regular pattern in the results, a pattern that
comes out clearly only after many repetitions.
Definition: We call a phenomenon random if individual
outcomes are uncertain but there is however a regular distribution
of outcomes in a large number of repetitions.
This remarkable fact is the basis for the idea of probability.
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The Language of Probability (Cont..)
Concept of Probability
1. Repeat an experiment (or observe a random phenomenon) a large
number of times.
2. Record the number of times a desirable outcome occurs.
3. Compute the ratio:
# of times the desirable outcome occurs
Total # of times the experiment was performed
Definition: The probability of any outcome of a random
phenomenon is the proportion of times the outcome would occur
in a very long series of repetitions.
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The Language of Probability (Cont..)
Example:
The statistics of a particular basketball player state that he makes 4
out of 5 free-throw attempts.
The basketball player is just about to attempt a free throw. What do
you estimate the probability that the player makes this next free throw
to be?
A. 0.16
B. 50-50. Either he makes it or he doesn’t.
C. 0.80
D. 1.2
Answer: C
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4.2 Probability Models
 Probability models
 Probability Rules
 Assigning Probabilities
 Independence and the Multiplication Rule
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Probability Models
Descriptions of chance behavior contain two parts:
1. List of possible outcomes and
2. A probability for each outcome.
 The set of all possible outcomes of a statistical experiment is called
the sample space and it is represented by the symbol S.
 An event is an outcome or a set of outcomes of a random
phenomenon. That is, an event is a subset of the sample space.
 A probability model is a description of some chance process that
consists of two parts: a sample space S and a probability for each
outcome.
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Probability Models
Example: Give a probability model for the chance
process of tossing of a coin.
Sample space , S = {Head, Tail}
Each of these outcomes has probability 1/2
Outcome
Probability
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Heads
1/2
Tails
1/2
Probability Models
Example: Give a probability model for the
chance process of tossing of a Die.
Each of these outcomes has probability 1/6
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Probability Models
Example: Give a probability model for the chance process of
rolling two fair, six-sided dice―one that’s red and one that’s green.
Sample Space
36 Outcomes
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Each outcome has probability 1/36.
Probability Rules
Rule 1. The probability of each outcome is a number between 0 and 1.
The probability P(A) of any event A satisfies 0 ≤ P(A) ≤ 1
Rule 2. The sum of the probabilities of all the outcomes in a sample
space equals 1.
If S is the sample space in a probability model, then P(S) = 1.
Rule 3. If two events A and B are mutually exclusive or disjoint
( A ∩ B = Φ, i.e., if A and B have no element in common) , then
P(A or B) = P(A U B ) = P(A) + P(B).
This is the addition rule for disjoint events.
Rule 4: The complement of any event A is the event that consists of
all the outcomes not in A, written AC.
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P(AC) = 1 – P(A)
Probability Rules
Example:
If you draw an M&M candy at random from a bag of the candies, the
candy you draw will have one of six colors. The probability of
drawing each color depends on the proportion of each color among all
candies made.
Assume the table below gives the probabilities for the color of a
randomly chosen M&M:
Color
Brown
Probability 0.3
Red
0.3
Yellow
?
Green
0.1
Orange
0.1
Blue
0.1
What is the probability of drawing a yellow candy?
Answer. 0.1
What is the probability of not drawing a red candy?
Answer: 1-0.3 = 0.7
What is the probability that you draw neither a brown nor a green candy?
Answer: 1-(0.3+0.1) = 0.6
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Probability Rules
Example:
Distance-learning courses are rapidly gaining popularity among
college students. Randomly select an undergraduate student who is
taking distance-learning courses for credit and record the student’s age.
Here is the probability model:
Age group (yr):
Probability:
18 to 23
0.57
24 to 29
0.17
30 to 39
0.14
(a) Show that this is a legitimate probability model.
Each probability is between 0 and 1 and
0.57 + 0.17 + 0.14 + 0.12 = 1
(b) Find the probability that the chosen student is not in the
traditional college age group (18 to 23 years).
P(not 18 to 23 years) = 1 – P(18 to 23 years)
= 1 – 0.57 = 0.43
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40 or over
0.12
Venn Diagrams
Sometimes it is helpful to draw a picture to display relations among several
events. A picture that shows the sample space S as a rectangular area and
events as areas within S is called a Venn diagram.
Two disjoint events:
Two events that are not disjoint, and
the event {A and B} consisting of
the outcomes they have in common:
A probability model with a finite sample space is called finite.
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Multiplication Rule for
Independent Events
If two events A and B do not influence each other, and if knowledge
about one does not change the probability of the other, the events are
said to be independent of each other.
Multiplication Rule for Independent Events
If A and B are independent:
P(A and B) = P(A) × P(B)
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