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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Larson & Farber, Elementary Statistics: Picturing the World, 3e 9

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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Larson & Farber, Elementary Statistics: Picturing the World, 3e 11

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

6Q8

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

Larson & Farber, Elementary Statistics: Picturing the World, 3e 20

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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Larson & Farber, Elementary Statistics: Picturing the World, 3e 30

Properties of Poisson Distribution

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Larson & Farber, Elementary Statistics: Picturing the World, 3e 31

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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Larson & Farber, Elementary Statistics: Picturing the World, 3e 32

Example 1

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Larson & Farber, Elementary Statistics: Picturing the World, 3e 33

Example 2

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Larson & Farber, Elementary Statistics: Picturing the World, 3e 34

Example 3

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Larson & Farber, Elementary Statistics: Picturing the World, 3e 35

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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Larson & Farber, Elementary Statistics: Picturing the World, 3e 39

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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Larson & Farber, Elementary Statistics: Picturing the World, 3e 41

The Normal Distribution: Graphically

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Larson & Farber, Elementary Statistics: Picturing the World, 3e 44

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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Larson & Farber, Elementary Statistics: Picturing the World, 3e 48

The Standard Normal Distribution (Z)

• ‘x’ is the data point in question.

• The mean (μ ) = 0

• Standard deviation (σ) =1