Chapter 2

Probability
(Definition, Binomial distribution, Normal distribution, Poisson’s distribution,
Properties)

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Probability
A probability is an action through which specific results
(counts, measurements or responses) are obtained.
Example:
Rolling a die and observing the
number that is rolled is a probability
experiment.
The result of a single trial in a probability experiment is
the outcome.
The set of all possible outcomes for an experiment is the
sample space.
Example:
The sample space when rolling a die has six outcomes.
{1, 2, 3, 4, 5, 6}www.DuloMix.com
Larson & Farber, Elementary Statistics: Picturing the World, 3e 2
Events
An event consists of one or more outcomes and is a subset of
the sample space.
Events are represented
by uppercase letters.
Example:
A die is rolled. Event A is rolling an even number.

A simple event is an event that consists of a single outcome.

Example:
A die is rolled. Event A is rolling an even number.
This is not a simple event because the outcomes of
event A are {2, 4, 6}.

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Larson & Farber, Elementary Statistics: Picturing the World, 3e 3
Types of Probability

1.Classical Probability
2. Empirical Probability
3. Subjective Probability

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Larson & Farber, Elementary Statistics: Picturing the World, 3e 4
Classical Probability
Classical (or theoretical) probability is used when each
outcome in a sample space is equally likely to occur. The
classical probability for event E is given by
Number of outcomes in event
P (E )  .
Total number of outcomes in sample space

Example:
A die is rolled.
Find the probability of Event A: rolling a 5.

There is one outcome in Event A: {5}
1
P(A) =  0.167
“Probability of 6
Event A.”
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Larson & Farber, Elementary Statistics: Picturing the World, 3e 5
Empirical Probability
Empirical (or statistical) probability is based on observations
obtained from probability experiments. The empirical
frequency of an event E is the relative frequency of event E.
P (E )  Frequency of Event E
Total frequency

f
n
Example:
A travel agent determines that in every 50 reservations
she makes, 12 will be for a cruise.
What is the probability that the next reservation she
makes will be for a cruise?
12
P(cruise) =  0.24
50 www.DuloMix.com
Larson & Farber, Elementary Statistics: Picturing the World, 3e 6
Law of Large Numbers
As an experiment is repeated over and over, the empirical
probability of an event approaches the theoretical (actual)
probability of the event.
Example:
Sally flips a coin 20 times and gets 3 heads. The
3
empirical probability is . This is not representative of
20
the theoretical probability which is 1 . As the number of
2
times Sally tosses the coin increases, the law of large
numbers indicates that the empirical probability will get
closer and closer to the theoretical probability.
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Larson & Farber, Elementary Statistics: Picturing the World, 3e 7
Probabilities with Frequency Distributions
Example:
The following frequency distribution represents the ages
of 30 students in a statistics class. What is the
probability that a student is between 26 and 33 years old?

Ages Frequency, f

18 – 25 13 P (age 26 to 33)  8
30
26 – 33 8
 0.267
34 – 41 4
42 – 49 3
50 – 57 2
 f  30 www.DuloMix.com
Larson & Farber, Elementary Statistics: Picturing the World, 3e 8
Subjective Probability
Subjective probability results from intuition, educated
guesses, and estimates.
Example:
A business analyst predicts that the probability of a
certain union going on strike is 0.15.

Range of Probabilities Rule
The probability of an event E is between 0 and 1,
inclusive. That is
0  P(A)  1.
Impossible 0.5 Certain
to occur Even to occur
chance
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Complementary Events
The complement of Event E is the set of all outcomes in
the sample space that are not included in event E.
(Denoted E′ and read “E prime.”)
P(E) + P (E′ ) = 1 P(E) = 1 – P (E′ ) P (E′ ) = 1 – P(E)

Example:
There are 5 red chips, 4 blue chips, and 6 white chips in
a basket. Find the probability of randomly selecting a
chip that is not blue.
4
P (selecting a blue chip)   0.267
15
4 11
P (not selecting a blue chip)  1    0.733
15 15
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Larson & Farber, Elementary Statistics: Picturing the World, 3e 10
Mutually Exclusive Events
Two events, A and B, are mutually exclusive if they
cannot occur at the same time.

A and B

A
B A B

A and B are A and B are not
mutually exclusive. mutually exclusive.
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Mutually Exclusive Events
Example:
Decide if the two events are mutually exclusive.
Event A: Roll a number less than 3 on a die.
Event B: Roll a 4 on a die.

A B
1
4
2

These events cannot happen at the same time, so
the events are mutually exclusive.
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Larson & Farber, Elementary Statistics: Picturing the World, 3e 12
Mutually Exclusive Events
Example:
Decide if the two events are mutually exclusive.
Event A: Select a Jack from a deck of cards.
Event B: Select a heart from a deck of cards.

A J 9 2 B
3 10
J J A 7
K 4
J 5
6Q8

Because the card can be a Jack and a heart at the
same time, the events are not mutually exclusive.
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Larson & Farber, Elementary Statistics: Picturing the World, 3e 13
Types of Probability Distribution
Discrete theoretical Continuous distribution
distribution
Binomial distribution Normal distribution
Poisson distribution Student’s t-distribution
Rectangular distribution Chi-Square distribution
Multinomial distribution F-distribution
Negative distribution
Geometric distribution
Hyper geometric
distribution
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Larson & Farber, Elementary Statistics: Picturing the World, 3e 14
Discrete Theoretical distribution
 The relationship between the values of a random variable
and the probability of their occurrence summarized by
means of a device called a probability distribution.
 It may be expressed in the form of a table, a graph or a
formula.
 Knowledge of probability distribution of a random variable
provides the researchers with a powerful tool for
summarizing and describing a set of data and for reaching
conclusion about a population on the basis of sample.
 Definition: The probability distribution of a discrete random
variable is a table, graph, formula, or other device used to
specify all possible values of a discrete random variable
along with their respective probabilities.
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Larson & Farber, Elementary Statistics: Picturing the World, 3e 15
Example
Prevalence of prescription and nonprescription drug use in
pregnancy among women delivered at a large eastern hospital.
N=4185
No. of 0 1 2 3 4 5 6 7 8 9 10
drugs
f 1425 1351 793 348 156 58 28 15 6 3 1

Construct a probability distribution table.
• What is the probability that a randomly selected women who
use 3 drugs?
• What is the probability that a woman picked at random who
used 2 or fewer drugs?
• What is the probability that a woman use 5 or more drugs?

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Larson & Farber, Elementary Statistics: Picturing the World, 3e 16
Solution
N=4185
No. of drugs f Probability
x
0 1425 1425 / 4185 = 0.340
1 1351 1351 / 4185 = 0.322
2 793 793 / 4185 = 0.189
3 348 348 / 4185 = 0.083
4 156 156 / 4185 = 0.037
5 58 58 / 4185 = 0.0138
6 28 28 / 4185 = 0.0066
7 15 15 / 4185 = 0.0035
8 6 6 / 4185 = 0.0014
9 3 3 / 4185 = 0.0007
10 1 1 / 4185 = 0.00023
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Larson & Farber, Elementary Statistics: Picturing the World, 3e 17
1. What is the probability that a randomly selected women who use 3 drugs?
• N= 4185
• Women who use 3 drugs = 348
• Probability = 348 / 4185 = 0.083

2. What is the probability that a woman picked at random who used 2 or
fewer drugs?
• N= 4185
• Women who use 2 or fewer drugs = 1425 + 1351+ 793 = 3569
• Probability = 3569 / 4185 = 0.852

3. What is the probability that a woman use 5 or more drugs?
• N= 4185
• Women who use 5 or more drugs = 58 + 28 + 15 + 6 + 3 + 1 = 111
• Probability = 111 / 4185 = 0.0265
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Larson & Farber, Elementary Statistics: Picturing the World, 3e 18
Binomial distribution

 Most widely encountered probability distribution.
 Derived from a process known as a Bernoulli trial.
 When a random process or experiment, called a trial can
result only one of two mutually exclusive outcomes, such
as success or failure, yes or no, true or false, dead or
alive, sick or well, male or female, the trial is called a
Bernoulli trial or Bernoulli experiment.

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Larson & Farber, Elementary Statistics: Picturing the World, 3e 19
Characteristics of a binomial random
variable
 The experiment consists of n identical trials.
 There are only 2 possible outcomes on each trial. We will
denote one outcome by S (for Success) and the other by F (for
Failure).
 The probability of S remains the same from trial to trial.
This probability will be denoted by p, and the probability of
F will be denoted by q ( q= 1-p).
 The trials are independent.
 Events are mutually exclusive (nothing is common between
two events, P(A U B) = P(A) + P(B) and exhaustive (When
events are complete in itself to form universe, P(A U B) = 1)
 The binomial random variable x OR r is the number of S’ in
n trials. www.DuloMix.com
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The probability distribution:

(r = 0, 1, 2, …, n),
Where,
p = probability of a success on a single trial,
q=1-p
n = number of trials, r = number of successes in n trials.

= combination of r from n

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Larson & Farber, Elementary Statistics: Picturing the World, 3e 21
Binomial Parameters
 The binomial distribution has two parameters n and p which
are sufficient to specify a binomial distribution.
 The mean of the binomial distribution is μ=np
 The variance of the binomial distribution σ2 = np(1-p)
(Variance is always less than mean and variance will be
highest when p=q)
 The standard deviation of the binomial distribution
σ = √npq
 If p = q = 0.5, the binomial distribution curve will be
symmetrical.
 When p<0.5, the binomial distribution is positively skewed.
 When p>0.5, the binomial distribution is negatively skewed.
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Values of the variable equidistant form their mean value.
i.e Mean = Median = Mode
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 The mode of the binomial distribution

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Example 1

0.0081 + 0.00045 + 0.00001 = 0.0086
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Example 2

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Example 3

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Example 4

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Larson & Farber, Elementary Statistics: Picturing the World, 3e 28
Poisson Distribution
 Poisson Distribution is a discrete probability
distribution and it is widely used in statistical
work . This distribution was developed by
French mathematician Dr. Simon Denis Poisson
in 1837 and the distribution is named after him.

 The Poisson Distribution is used in those
situations where the probability of happening of
an event is small ,i.e. the event rarely occurs.

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Larson & Farber, Elementary Statistics: Picturing the World, 3e 29
Poisson Distribution is defined and given by the
following probability function:

Where,
• P(r) = probability of obtaining r number of
success.
• m = np = parameter of distribution .
• e = 2.7183 base of natural logarithms
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Properties of Poisson Distribution

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Applications
 It is used in statistical quality control to count the number
of defects of an item.
 In Biology, to count the number of bacteria.
 In insurance problems to count the number of causalities.
 To count the number of errors per page in a typed material.
 To count the number of incoming telephone calls in a town.
 To count the number of defective blades in a lot of
manufactured blades in a factory.
 To count the number of deaths at a particular crossing in a
town as a result of road accident.
 To count the number of suicides committed at lover point
in a year
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Example 1

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Example 2

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Example 3

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Example 4

Between 9am and 10am, average
number of phone calls per minute
coming is 4.Find the probability that
during one particular minute, there will
be,
(a) No phone calls
(b) At most 3 phone calls

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Larson & Farber, Elementary Statistics: Picturing the World, 3e 36
Example 5
If 2% of electric bulbs manufactured by a
company are known to be defectives, what
is the probability that a sample of 150
bulbs taken from production process of
that company would contain
(a) Exactly one defective bulb
(b) More than 2 defective bulb.

0.15
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Larson & Farber, Elementary Statistics: Picturing the World, 3e 37
Normal Distribution
 It is defined as a continuous frequency distribution of
infinite range.
 The normal distribution is a descriptive model that describes
real world situations.
 IMPORTANCE:
• Many dependent variables are commonly assumed to be
normally distributed in the population.
• If a variable is approximately normally distributed we can
make inferences about values of that variable.
 It is sometimes called the “bell curve,”.
 It is also called the “Gaussian curve” after the
mathematician Karl Friedrich Gauss.

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Larson & Farber, Elementary Statistics: Picturing the World, 3e 38
Symbols Used
 “z” – z-scores or the standard scores. The table
that transforms every normal distribution to a
distribution with mean 0 and standard deviation
1. This distribution is called the standard normal
distribution or simply standard distribution and
the individual values are called standard scores
or the z-scores.
 “μ” – the Greek letter “mu,” which is the Mean,
 “σ” – the Greek letter “sigma,” which is the
Standard Deviation.

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Features of Normal Distribution

1. Normal distributions are symmetric around their mean.
2. The mean, median, and mode of a normal distribution are
equal.
3. The area under the normal curve is equal to 1.0, half the
area under the normal curve is to the right of this center point
and the other half to the left of it.
4. Normal distributions are denser in the center and less
dense in the tails. It is Asymptotic: The curve gets closer and
closer to the X –axis but never actually touches it. To put it
another way, the tails of the curve extend indefinitely in both
directions.
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Larson & Farber, Elementary Statistics: Picturing the World, 3e 40
5. Normal distributions are defined by two parameters, the
mean (μ) and the standard deviation (σ).
6. 68% of the area of a normal distribution is within one
standard deviation of the mean.
7. Approximately 95% of the area of a normal distribution is
within two standard deviations of the mean.

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The Normal Distribution: Graphically

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What Is Skewness in Statistics?
 Some distributions of data, such as the bell curve are
symmetric. This means that the right and the left of the
distribution are perfect mirror images of one another.
 Not every distribution of data is symmetric. Sets of data that
are not symmetric are said to be asymmetric.
 The measure of how asymmetric a distribution can be is
called skewness.
 The mean, median and mode are all measures of the center
of a set of data.
 The skewness of the data can be determined by how these
quantities are related to one another.
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Larson & Farber, Elementary Statistics: Picturing the World, 3e 45
SKEWED TO THE RIGHT
 Data that are skewed to the right have a long tail that
extends to the right. An alternate way of talking about a
data set skewed to the right is to say that it is positively
skewed. In this situation the mean and the median are both
greater than the mode. As a general rule, most of the time
for data skewed to the right, the mean will be greater than
the median.
1. Always: mean greater than mode
2. Always: median greater than mode
3. Most of the time: mean greater than median

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Larson & Farber, Elementary Statistics: Picturing the World, 3e 46
SKEWED TO THE LEFT
 The situation reverses itself when we deal with data skewed
to the left. Data that are skewed to the left have a long tail
that extends to the left. An alternate way of talking about a
data set skewed to the left is to say that it is negatively
skewed. In this situation the mean and the median are both
less than the mode. As a general rule, most of the time for
data skewed to the left, the mean will be less than the
median.
1. Always: mean less than mode
2. Always: median less than mode
3. Most of the time: mean less than median

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Larson & Farber, Elementary Statistics: Picturing the World, 3e 47
The Normal Distribution: as mathematical function

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The Standard Normal Distribution (Z)

• ‘x’ is the data point in question.
• The mean (μ ) = 0
• Standard deviation (σ) =1