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Business Statistics - QBM117

Assigning probabilities to events

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Objectives

To define probability;

To describe the relationship between randomness and probability;

To define a random experiment;

To revise the methods of assigning probabilities;

To introduce some of the probability rules.

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What is probability?

Probability is a numerical measure of uncertainty;

It is a number that conveys the strength of our belief in the occurrence of an uncertain event;

It is often associated with gambling;

It is now an indispensable tool in the analysis of situations which involve uncertainty.

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Randomness

What is the mean age of all the students in this class?

From a random sample of 10 students we can estimate the mean age of all the students.

How can the mean, based on only a sample of 10, be an accurate estimate of the population mean, ?

A second random sample would most likely produce a different value for the mean.

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This is due to sampling variability.

Why is this not a problem?

Chance behaviour is unpredictable in the short term but has a regular and predictable pattern in the long term.

For example, consider the experiment of tossing a coin

The results cannot be predicted in advance but there is a pattern which emerges, only after repeated sampling.

This is the basis for probability.

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Randomness and probability

A phenomenon is called random if the individual outcomes are uncertain, but there is a regular distribution of outcomes in a large number of repetitions.

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.

Probability is an idealisation based on what would happen in an infinitely long series of trials.

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Random experimentsWe begin our study of probability by considering the random experiment, as this process generates the uncertain outcomes to which we assign probabilities.

A random experiment is any well defined procedure that results in one of a number of possible outcomes.

The outcome that occurs, cannot be predicted with certainty.

For example, rolling a die and observing the number uppermost on the die.

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Important

The actual outcome of a random experiment cannot be determined in advance.

We can only talk about the probability that a particular outcome will occur.

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Example

If we roll a die, the uppermost face can be a

1, 2, 3, 4, 5 or a 6. These numbers form the

sample space.

Sample Space

1

3

4

5

6

2

is the complete list of all the possible outcomes of an experiment.

The sample space

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Example

Rolling a die with an even number

uppermost can be an event.

Sample Space

1 3 5

2 4 6

Events

The desired outcome or outcomes from the sample space.

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Example

In a family of two children the

event of there being at least

one boy is:

Sample Space

girl / girl boy / boy

girl / boy

boy / girl

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Example

If we roll a die the uppermost face can be

either a 1, 2, 3, 4, 5 or a 6.

These numbers form the sample space.

The sample space can be written as:

6 5, 4, 3, 2, 1, S

Notation

Events can be described in written form, provided the label is defined before hand.

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2 1, B

Example

When rolling a die, let A

equal the even numbers.

6 4, 2, A

When rolling a die, let B

equal the numbers less than three.

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The probability of an event is defined as the number of members in an event divided by the number of members in the sample space.

Probability of an event

SnAn

AP

Where n(A) is the number of members in the event A and n (S ) is the number of members in the sample space.

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Example

When rolling a die:

, , , , , S 654321

6 4, 2, A {The event of an even number}

5.063

SnAn

AP

The probability of A is:

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Example

On a particular day a statistics lecture has 260 students attending. There are 65 mature aged students in the lecture.

If one student is selected at random, what is the probability that a mature age student is selected?

25.026065

SnMn

MP

S ={all the students in the lecture}

M = {mature age students in the lecture}

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A~ cAAAA

The complement of an event are those members of the sample space that are not contained within the event.

Complement of an event

Complement rule

APAP 1

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Example

When rolling a die, event A is defined

as the even numbers. Find :ASample Space

A2 4 6

1 3 5 A

5.06

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)(1)(

APAP

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The intersection of two events are the members that are common to both events.

The word used to represent the intersection of two events is:

AND

Intersection

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Example

When rolling a die, let A

be the event an even number

is uppermost. 6 4, 2, A

When rolling a die, let B be the event a

number less than three is uppermost.

2 1, B

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Sample Space

Find the intersection of A and B.

1 3 5

2 4 6

It’s just me, all alone…

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Find the probability of A and B.

1667.06

1

and and

Sn

BAnBAP

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Example

Let A be defined as the students with2 blue eyes in a lecture theatre and B be defined as the students with two browneyes in the lecture theatre.

Then the probability students will have two blue and two brown eyes in the lecture theatre is ...

Put your hand up if you

look like me!

0 and BAP

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Two events are mutually exclusive when they have no members in common. (They don’t share any members)

Mutually exclusive events

0 and

0) and(

BAP

BAn

A B

Example

A is the event an even number is uppermost on a die

B is the event an odd number is uppermost on die

2 4 6 1 3 5

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The union of two events are those members that are in one event or the other event or in both.

The word used to represent the union of two events is:

OR

Union of events

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Example

When rolling a die, let X be the

event an odd number is uppermost.

5 3, 1, X

When rolling a die, let Y be the event

a number less than five is uppermost.

2,3,4 1, Y

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Sample Space

Find the probability of X or Y.

1 3 5

2 4 6

All except for me !!

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Therefore the probability of X or Y is

8333.06

5

or or

Sn

YXnYXP

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We do not always know the members of each event,rather we only know the probabilities of these events.

In such cases, there is a formula that enables us to findthe probability of A or B:

BAPBPAPBAP andor

Addition formula

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Sample Space

A BACounted once

with P(A) BCounted twice

with P(B)

Intersection

Addition formula

BAPBPAPBAP andor

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Example

In a population the probability of being female is 0.6 and the probability of being aged 30 and over is 0.4. The probability of being female and aged 30 and over is 0.2. Find the probability of being either female or aged 30 and over.

Let F represent the females in the populationLet O represent the people aged 30 and over

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Example

In a population the probability of being female is 0.6 and the probability of being aged 30 and over is 0.4. The probability of being female and aged 30 and over is 0.2. Find the probability of being either female or aged 30 and over.

6.0FP 40.OP 2.0and OFP

8.0

2.04.06.0

andor

OFPOPFPOFP

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Reading for next lecture

Chapter 4 sections 4.4 - 4.5

Exercise to be completed before next lecture

S&S 4.5 4.7 4.13 4.15 4.69