Lecture 5-Probability

Question 1

What is Probability

Answer 1

is the study of random or non-deterministic situations.

Question 2

Explaining Probability

Answer 2

Underlying statements such as these is at least some vague idea about the likelihood that there will be a particular outcome (he will drop by, it will etc.). If the person making the statement is asked “How likely?”, or better yet “How probable, on a scale of 0 to 10?”, an answer, say “6” or “7” may be given. The closer the value to 10, the more likely is the possible outcome and, in the same vein, the closer the value to 0, the less likely. Conventionally, it is the scale 0-1 that is used (or 0% to 100%). The extreme values, 0 and 1, are, in the strict English sense of the word, not really probabilities. The lower limit, 0, represents the certainty that the outcome will not occur while the upper limit, 1, represents the absolute certainty that the outcome will occur.
*It is the in-between values which indicate the existence of doubt about the occurrence of the outcome and at the same time measures the strength or weakness of the doubt.

Question 3

What is the probability of an event?

Answer 3

a numerical measure of the likelihood that the event will occur.

Question 4

When does the probability of an even occur?

Answer 4

Gambling decisions
*Insurance premiums
*Inheritance of genetic characteristics

Question 5

What is sample space?

Answer 5

A collection of all possible outcomes of an experiment is known

Question 6

Example 1 of Sample Space

Answer 6

Suppose the experiment consisted of flipping a fair coin. The flip could only result in either the coin landing Head up or landing Tail up.
*In short, we say that the possible outcomes of this experiment are Head and Tail.
*Thus the sample space will comprise the two outcomes of Head and Tail.
*It is customary to write this as a set in which the designation for sample space is the capital letter S. Hence we write S = { Head , Tail }.

Question 7

Example 2 of Sample Space

Answer 7

Suppose the experiment consisted of interviewing first year students to measure their level of satisfaction with the registration process. To do so, we may ask some or all the students the following question:
“Which of the following statements best describes your level of satisfaction with the recently concluded registration process:
Very Dissatisfied Dissatisfied Neutral Satisfied Very Satisfied.”
*The possible outcomes of this experiment will be responses - Very Dissatisfied, Dissatisfied, Neutral, Satisfied, and Very Satisfied. These will therefore constitute the sample space.
*Hence we write
S = {Very Dissatisfied, Dissatisfied, Neutral, Satisfied, Very Satisfied}.

Question 8

What is a Venn Diagram?

Answer 8

closed shape (box, rectangle, circle etc.) that depicts all the possible outcomes for an experiment.

Question 9

What is a subset of a sample space?

Answer 9

An event. It can be simple or compound.

Question 10

What is a simple event

Answer 10

comprises only one outcome from the sample space.
*For example in the experiment of rolling a fair
*die, the Sample Space is given by
S = {1, 2, 3, 4, 5 6}.
*Each outcome in this sample space is a simple event. Thus there are 6 simple events here.
{1} {2} {3} {4} {5} {6}

Question 11

What is a compound event?

Answer 11

comprises more than one outcome.
For example in the experiment of rolling a fair die,
S = {1, 2, 3, 4, 5, 6 }
There are also:
* compound events comprised of two outcomes e.g. the event {1 , 2};
* compound events comprised of three outcomes e.g. the
event { 2, 4, 6};
*compound events comprising four outcomes e.g. the
event {1, 2 3, 4}; and
* compound events comprising five outcomes e.g. the event {1, 2, 3, 4, 5}.

Question 12

Three approaches to determining the probability of an event

Answer 12

The Classical Approach
The Relative Frequency Approach
The Subjective Approach
*These approaches span the continuum from objectivity to subjectivity.

Question 13

Definition of classical approach

Answer 13

If A is a simple event from a sample space S of a
given experiment, the Classical approach
determines the probability of A by the formula
1
P(A) =
Number of Simple Events in the Sample Space S

Question 14

Examples of the Classical Approach

Answer 14

Consider the toss of a ‘fair’ coin, the Sample Space is {Head, Tail}. By the ‘fairness’ of the coin, the outcomes of Head and Tail are equally likely to occur. Thus
P(Head) = ½ and P(Tail) = ½.
*Consider the roll of a ‘fair’ die, the Sample Space is { 1, 2, 3, 4, 5, 6 }. By the ‘fairness’ of the die, the outcomes of 1, 2, 3, 4, 5, and 6 are equally likely to occur. Thus
P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6.
In both these examples, the probabilities were clear once we knew
how many simple events comprised the sample space.

Question 15

What is priori probabilities?

Answer 15

Probabilities determined by the classical approach

Question 16

Generalization

Answer 16

P(E)= number of favorable outcomes over total number of possible outcomes
Example:
Suppose there are 22 females and 8 males in a group. What is the probability of randomly selecting a female from the
group?

P(female)= 22 over 30= 30 over 15
or

Question 17

Limitation of Classical Approach

Answer 17

There is one shortcoming of the Classical Approach and it is that few situations, if any, in the real world qualify as giving rise to equally likely outcomes. As such, it is not practical for several situations found in the real world.
*Consider for example the probability that a student passes a course like ECON1005. The sample space here is comprised of Pass and Fail. However, the pass rate is not 50%; thus the events are not equally likely.
*We therefore need an approach that can be applied to most situations in the real world.

Question 18

Defining Relative Frequency Approach

Answer 18

If an experiment is repeated a sufficiently large number of times, the relative frequency of a particular event equals the probability of that event.
Typically, in the real world, reasons of cost etc. sometimes make it impossible to repeat an experiment a very large number of times. Hence some refinement is necessary;
*If an experiment is repeated N times and an event B occurs in n of these times, then the probability of event B is given by
P(B) = n/N

Question 19

Long explanation of Relative Frequency Approach

Answer 19

Suppose that we wanted to compute any of the following probabilities:
*The probability that a family selected at random owns two cars
*The probability that an 80-yr old person will live for at least one more year
The underlying experiments for these situations are as follows:
–The experiment of selecting a family at random and recording whether the family owns 2 cars or not;
–The experiment of selecting an 80-year old person today and checking in one year’s time whether that person is still alive;

The Classical Approach will not be applicable in any of these experiments since the outcomes for these experiments are not equally likely.
Instead we recognize that:
*each of these experiments can be performed repeatedly;
*after a large number of repetitions we can compute the frequency of occurrence of the given event and hence the relative frequency of the given event.

Question 20

Generalization (relative frequency ?)

Answer 20

number of times an event occurred in the past
P(E) over
total number of opportunities of the event to occur
Example:
Data collected from your company record books show that the supplier had sent your company 80 batches in the past, and inspectors had rejected 15 of them. By the method of
relative probability, the probability of the inspectors rejecting the next batch is 15/80, or 0.19.

Question 21

Limitations of Relative Frequency Approach

Answer 21

Both the Classical and Relative Frequency Approaches are considered to be objective approaches.
*There are however experiments in which the equally likely property of the outcomes in the sample space is not present and/or it is impractical to generate a large number of repetitions due to cost considerations, the destructive nature of the experiment, or little or no history of such experiments.
* In such situations neither the Classical nor the Relative Frequency Approach will be applicable; we must look to the only remaining approach which is subjective by nature.

Question 22

Define the Subjective Approach

Answer 22

is an approach in which the probability of an event is assigned on the basis of subjective judgement, belief, experience and/or information. The assignment is arbitrary and influenced by the biases, preferences and experience of the person(s) assigning the probability.

Question 23

Example of the subjective approach

Answer 23

The probability that students who attend their lectures and tutorials will earn an A in Econ 1005
*The probability that West Indies will their next one-day match
*The probability that it will rain tomorrow
Conceptually:
*Medical doctors sometimes assign subjective probabilities to the length of life expectancy for people having cancer.
*Weather forecasting is another example of subjective probability.

Question 24

Comparing Approaches Pt 1

Answer 24

The philosophies underlying the two approaches appear to be quite distinct.
*The Relative Frequency Approach assigns probabilities to outcomes based on the proportion of times the outcome occurs when an experiment is repeated an unlimited number of times
*Take, the example of the sun rising in the morning. We know that for all past repetitions of the experiment (that is, for every new day), the outcome (the sun rises) is observed, and so we assign to this outcome a probability of one.
*The Classical Approach imposes a probability because of some prior understanding of the phenomenon under study; for example, finding the likelihood of drawing a male or a female when randomly drawing a student from the Faculty.

Question 25

Comparing Approaches Pt 2

Answer 25

In the final analysis, the two approaches do not fundamentally differ.
*Take for example, the coin toss. Since we know that the options are only head or tail, we assume that the probability of obtaining one versus the other is a half (the Classical Approach).
*If we were to perform this experiment (the coin toss) an infinite number of times, however, we would find that approximately one half of the times we would obtain a head (the Relative Frequency Approach).
*To a large extent, it becomes difficult to distinguish one approach from the other. In any event, lack of knowledge with respect to the probability approach does not preclude us from imposing a probability.
*The challenge is, therefore, on attaching the probability to the outcome in question, rather than discussion on the genesis of the probability measure.

Question 26

Mutually Exclusive Events

Answer 26

Two events B and C from the same sample space are said to be mutually exclusive events if they have no outcomes in common.
*In other words, the intersection of B and C is the null event.
*Hence the P(B and C ) = P(Ø) = 0
*Activity: how will mutually exclusive events be drawn in a Venn Diagram?

Question 27

Define Union of Events

Answer 27

is the event that either B alone occurs or C alone occurs or both B and C occur. It is usually referred to as B or C.

Question 28

What is the addition rule

Answer 28

Method used to calculate the union of events

Question 29

Define Complement of an Event

Answer 29

*Given an event B in a sample space, the complement of an event B (written Bc) is the event that B does not occur.
Hence P(Bc) = P( event B does not occur ).

Question 30

Define Null of an Event

Answer 30

The null event Ø is the event which comprises no outcomes from the sample space.
Put differently, it is the event that never occurs.
Hence P( Ø) = 0.

Question 31

Define Intersection of an events

Answer 31

The intersection of two events B and C is the event that both B and C occur. It is usually referred to as B and C.
*The probability of the intersection of two events is their joint probability.
*The method used to calculate this probability is the Multiplication Rule. We will come to this later.

Question 32

Independent Events

Answer 32

Two events are said to be independent if the occurrence of one does not affect the probability of the occurrence of the other
–P(B | C) = P( B) and P( C | B) = P(C).

Question 33

Dependent events

Answer 33

f the occurrence of one event affects the probability of the occurrence of the other event, then the two events are said to be dependent events

Question 34

Tossing a Die pt 1

Answer 34

The trial (the die is actually tossed) of an experiment (tossing the die) can result in a particular outcome (one of the six numbered faces).
*The number of times an outcome is observed (a particular face appears) relative to the number of trials of the experiment seems to be a reasonable measure of the probability of the outcome.
*If the die is fair (we say that it is not loaded), your intuition that the probability of a one appearing is equal to one sixth ought to be borne out by the probability assigned using the relative frequency approach.

Question 35

Tossing a die pt 2

Answer 35

An experiment is an action that can be repeated. Here, the experiment is the toss of a die.
*There may be (theoretically or practically) several possible trials of such an experiment - here, the act of tossing the die.
*For each trial, the same possible outcomes must always be possible - here, whether we obtain a one, a two (or any of the other faces) when the die is tossed.
*At the end, the experiment may be repeated, and the same outcomes may or may not be observed. A sample space is a complete enumeration or description of all the likely mutually exclusive outcomes associated with a trial of an experiment.
*The sample space of this experiment would be defined as {F1, F2, F3, F4, F5, F6}
*When the die is thrown, no more than one of these outcomes is possible i.e. they are mutually exclusive.

Question 36

The addition law of probability

Answer 36

states that given any two events B and C from a sample space,
P(B or C) = P(B) + P(C) - P(B and C)
*Note that the operative word here is ‘OR’.

Question 37

Special case of addition law of probability

Answer 37

states that given any two mutually exclusive events B and C from a sample space,
P(B or C) = P(B) + P(C)
*In other words, P(B and C) = 0

Question 38

Contingency Tables

Answer 38

The responses of 100 people in TT to whether or not they are in favour of or against the death penalty
In Favour Against Total
Male 15 45 60
Female 4 36 40
Total 19 81 100

*This Table shows the distribution of the respondents based on gender and opinion.

Question 39

Marginal Probability/Simple Probability

Answer 39

is the probability of a single event without consideration of any other event.

If only one characteristic is considered at a time (male, female, in favour of, against), the probability of each of these 4 characteristics or events

Question 40

What is Conditional Probability? (short)

Answer 40

restricts our attention to the event C that has already occurred. As such we treat the event C as a ‘restricted’ sample space. The probability that event B will occurs in the ‘restricted’ sample space is the conditional probability.
given that event (male) has already happened
P( in favour GIVEN that the person is male)
*P(in favour | male)
Conditional probabilities are always recognized by phrases such as ‘given that’, ‘in light of’, ‘in view of’ etc.

Question 41

Conditional Probability Explanation

Answer 41

Now, suppose that one employee is selected at random from these 100 people, and assume that we know that the person selected is Male
*In other words, the event that the person selected is a Male has already occurred

Question 42

Example of Conditional Probability

Answer 42

In Favour Against Total
Male 15 45 60
Female 4 36 40
Total 19 81 100
Probability (in favour GIVEN male)
–Total sample = 100
–Restricted sample space = males = 60
–In this sample space, there are 15 males in favour
–Therefore, probability (in favour GIVEN male) = 15/60
–This is the same as dividing 15/100 by 60/100

Question 43

Conditional Probability/ Multiplication Law of Probability

Answer 43

Given two events B and C of the same sample space, the conditional probability of B given C, interpreted i.e. the probability that an event B will occur given that another event C has already occurred, and written P(B | C) , is given by the formula
P(B C) over
P(B | C) = P(C)

In the example,
B = in favour of, C = Male, P(B and C) = 15/100 and P(C) = 60/100. We therefore compute P(B | C) as 15/60
*If we re-arrange this, we can calculate
P(B C) = P(B | C) * P(C)

The multiplication law of probability states that given any two events B and C from the same sample space,
P(B and C) = P(B|C) x P(C).
*Alternatively
P(B and C) = P(C |B) x P(B).
*Note here that the operative word is ‘and’.

Question 44

Example of Multiplication Law/Rule

Answer 44

Suppose there are 10 marbles in a bag, and 3 are defective. Two marbles are to be selected, one after the other without replacement.
*What is the probability of selecting a defective marble followed by another defective marble?
*A = first marble selected is defective
*B = second marble selected is defective
*We need to find P(A AND B) = P(BA) * P(A)
*P(A)=3/10
*Probability that the second marble selected is defective, given that the first marble is defective: P(B GIVEN A) = P(BA) =2/9
*Therefore, P(A and B) = (3/10) (2/9) = 7%

Question 45

Special Case of the Multiplication Law

Answer 45

The Multiplication Law: P(B and C) = P(B|C) x P(C)
*Do you remember how we defined an independent event?
*Two events are said to be independent if the occurrence of one does not affect the probability of the occurrence of the other, for example, P(B | C) = P( B)
*Therefore, the special case of the multiplication law of probability states that given any two independent events B and C from the same sample space,
P(B and C) = P(B) x P(C).

Question 46

The Axioms of Probability

Answer 46

The probability of every event is at least zero: For every event A, P(A) ≥ 0. (There is no such thing as a negative probability.)
*The probability of the entire outcome space is 1: P(S) = 1. (The chance that something in the outcome space occurs is 100%, because the outcome space contains every possible outcome.)
*If two events are disjoint, the probability that either of the events happens is the sum of the probabilities that each happens. (If AB = {}, P(A ∪ B) = P(A) + P(B).)