A Survey of Probability Concepts

Lecture 3

Define probability. Explain the terms experiment, event and

outcome. Describe the classical, empirical, and

subjective approaches to probability. Calculate probabilities using the rules of

addition, complement rule, joint probability, rules of multiplication and conditional probability.

Apply contingency table and tree diagram to organize and compute probabilities.

Bayes’ theorem. Explain and apply multiplication,

permutations, and combinations.

GOALS

A probability is a measure of the likelihood that an event in the future will happen. It it can only assume a value between 0 and 1.

A value near zero means the event is not likely to happen. A value near one means it is likely.

Definitions

An experiment is the observation of some activity or the act of taking some measurement.

An outcome is the particular result of an experiment.

An event is the collection of one or more outcomes of an experiment.

Definitions

Tossing coins Rolling dices

Experiments

Outcomes 2 x ‘Head’1x‘Tail’ + 1x‘Head’1x‘Head’ + 1x‘Tail’2 x ‘Tail’

2 x ‘6’1x‘5’ + 1x‘6’… …1x‘1’ + 1x‘2’2 x ‘1’

Events Observe 2 x ‘Head’Observe 2 x (1x‘Head’+ 1x‘Tail’)

Observe 3 outcomes whereby the sum of both die is more than 10

Experiments, Events and Outcomes

There are three approaches of assigning probability:◦classical, ◦empirical, ◦subjective.

Assigning Probabilities

Classical ProbabilityConsider an experiment of rolling two six-sided die.- What is the probability of the event “the sum of both

die is more than 10”?- P(sum of both die > 10) = 3/36

The possible outcomes are:*Students to fill in the 36 possible outcomes as

homework.

There are 3 possible outcomes in the collection of 36 equally likely possible outcomes.

outcomes possible ofnumber Total

outcomes favorable ofNumber eventan ofy Probabilit

YPROBABILIT CLASSICAL

Events are mutually exclusive if the occurrence of any one event means that none of the others can occur at the same time.

Events are independent if the occurrence of one event does not affect the occurrence of another.

Events are collectively exhaustive if at least one of the events must occur when an experiment is conducted.

Mutually Exclusive, Independent, Collectively Exhaustive

Empirical Probability:The probability of an event happening is the fraction of the time similar events happened in the past.

◦ The key to establishing probabilities empirically is that more observations will provide a more accurate estimate of the probability.

◦ The empirical approach to probability is based on what is called the law of large numbers.

Empirical Probability

Law of Large Numbers:Over a large number of trials the empirical probability of an event will approach its true probability.

Law of Large Numbers Suppose we toss a fair die.

The result of each roll is either ‘1’, ‘2’, ‘3’, ‘4’, ‘5’ or ‘6’.

If we roll the die a great number of times, the probability of each outcome will approach 1/6.

The following table reports the results of an experiment of rolling a fair die 1, 10, 50, 100, 500, 1,000, 5000 and 10,000 times and then computing the relative frequency of ‘1’.

Note that this result is not repeatable. Why?*Try repeating the experiment for ‘2’, up to 100 times.

Number of

Trials

Number of '1'

Relative Frequency

of '1'

1 0 0.00000

10 2 0.20000

50 8 0.16000

100 17 0.17000

500 82 0.16400

1000 165 0.16500

5000 850 0.17000

10000 1655 0.16550

In 2006, there were floods experienced in parts of Singapore, after exceptional heavy rainfall. This was the second time in past 90 rainy days. On the basis of this information, what is the probability that a future rainy day would cause floods?

Empirical Probability

022.090

2

daysrainy ofnumber Total

daysrainy during flood ofNumber flood a ofy Probabilit

Subjective Concept of Probability:The probability of a particular event happening that is assigned by an individual based on whatever information is available.

If there is little or no past experience or information on which to base a probability, it may be arrived at subjectively.

Examples of subjective probability are:1. Estimating the likelihood the Singapore Soccer Team

make it to World Cup.2. Estimating the likelihood that your neighbour will be

married before the age of 30.3. Estimating the likelihood the Singapore budget

surplus would exceed $5B next year.

Subjective Probability

Summary of Types of Probability

Approaches to Probability

Subjective Objective

Classical Probability Empirical Probability

Based on equally likely outcomes

Based on relative frequencies

Based on available information

Special Rule of Addition If two events A and B are mutually exclusive, the probability of one or the other event’s occurring equals the sum of their probabilities. P(A or B) = P(A) + P(B)

General Rule of AdditionIf A and B are two events that are not mutually exclusive, then P(A or B) is given by the following formula:P(A or B) = P(A) + P(B) - P(A and B)

Probability: Rules of Addition

Event A Event B

Event A Event B

P(A and B)

Addition Rule What is the probability that a card chosen at

random from a standard deck of cards will be either a queen or a club?

P(A or B) = P(A) + P(B) - P(A and B) = 4/52 + 13/52 - 1/52 = 16/52, or .3077

Card Probability Explanation

Queen P(A) = 4/52 4 queens out of 52 cards

Club P(B) = 13/52 13 clubs out of 52 cards

Queen of Club P(A and B) = 1/521 queen of clubs out of 52

cards

Club Queen

Queen of Club

The complement rule is used to determine the probability of an event occurring by subtracting the probability of the event not occurring from 1.

P(A) + P(A’ ) = 1 or P(A) = 1 - P(A’ ).

Complement Rule

A

A’

JOINT PROBABILITY: Probability that two or more events will happen concurrently.

Joint Probability

Club Queen

Queen of Club

Special rule of multiplication1. Two events A and B are independent. 2. Two events A and B are independent if the occurrence of one has no effect on the probability of the occurrence of the other.

P(A and B) = P(A).P(B)

Probability: Rule of Multiplication

Conditional probability is the probability of a particular event (A) occurring, given that another event (B) has occurred.

When Event A occurs after Event B, and Event B has an effect on the likelihood that Event A occurs, then A and B are dependent.

The probability of Event A given that Event B has occurred is written P(A|B).

Conditional Probability

General Multiplication Rule The general rule of multiplication is used to

find the joint probability that two events will occur, when the events are not independent.

It states that for two events, A and B, the joint probability that both events happening is found by multiplying the probability that Event A, P(A), will happen by the conditional probability of Event B occurring given that A, P(B|A), has occurred.

A)|P(A).P(BB) andP(A

:tionMultiplica of Rule General

There are 12 coloured balls in a bag. 9 of these balls are red and the others green. If we would to draw two balls consectively, without replacement. What is the likelihood both balls selected are red?

General Multiplication Rule

• The event that the first ball selected is red is R1. The probability is P(R1) = 9/12

• The event that the second ball selected is also red is identified as R2. The conditional probability that the second ball selected is red, given that the first shirt selected is also red, is P(R2 | R1) = 8/11.

• To determine the probability of 2 red ball being selected we use formula: P(AnB) = P(A).P(B|A)

• P(R1 and R2)

= P(R1).P(R2 |R1) = (9/12)(8/11) = 0.55

Contingency TablesContingency Table used to classify sample observations according to two or more identifiable characteristics.

Number of Credit Cards0

(B1)1

(B2)2

(B3)3 or more

(B4) Total

Men (A1) 2 4 3 3 12

Women (A2) 4 3 6 5 18

Total 6 7 9 8 30

What is the probability of a randomly selected person is a women and does not have any credit cards?• Event B1 happens if a randomly selected person does not have any

credit cards. P(B1) = 6/30, or 0.2.• Event A2 happens if a randomly selected person is a women. P(A2) = 18/30, or 0.6.• Of the 18 women, 4 do not have credit cards. P(B1 | A2) is the

conditional probability that women do not have any credit cards, • P(B1 |A2) = 4/18• P(A2 and B1) = P(A2).P(B1 |A2) = 18/30 *4/18 =

4/30

A tree diagram is useful for illustrating conditional and joint probabilities. It is particularly useful for analyzing decisions and organizing calculations involving several stages. Each segment is one stage of the problem. The branches of a tree diagram are weighted probabilities.

Tree Diagrams

Tree Diagram

Men

Women

1 credit card

2 credit card

>3 credit card

0 credit card

1 credit card

2 credit card

>3 credit card

0 credit card

Conditional Probabilities

2/12

3/123/12

4/12

4/18

3/18

6/185/18

Gender

CreditCard

JointProbabilities

12/30

18/30

12/30 * 2/12 = 2/30

12/30 * 4/12 = 4/30

12/30 * 3/12 = 3/30

12/30 * 3/12 = 3/30

18/30 * 4/18 = 4/30

18/30 * 3/18 = 3/30

18/30 * 6/18 = 6/30

18/30 * 5/18 = 5/30

Must Be Equal to 1.00

Read textbook pages 158 and 159.

Bayes’ Theorem

Bayes' theorem relates the conditional and prior probabilities of random events.

Prior Probability: The initial probability based on the present level of information.

Posterior Probability: A revised probability based on additional information.

An)|P(An).P(B...A2)|P(A2).P(B A1)|P(A1).P(B

Ai)|P(Ai).P(BB)|P(Ai

WHO suspected that 5% of Country A’s population are infected with flu.◦ A1: Event that person has flu

P(A1) = 0.05◦ A2: Event that person does not have flu

P(A2) = 1-P(A1) = 0.95 There is a diagnostic test available to detect the disease, but its

accuracy is 90%.◦ Event that person has flu and detected by test

P(B|A1) = 0.9◦ Event that person does not has flu, but detected by test

P(B|A2) = 0.15

When we randomly select a person and he is tested positive. What is the probability the person actually has flu?◦ Find, P(person has flu | test is positive) ◦ Symbolically, P(A1|B)

24.015.0*95.09.0*05.0

9.0*05.0

A2)|P(A2).P(B A1)|P(A1).P(B

A1)|P(A1).P(BB)|P(A1

Bayes’ Theorem

*Read textbook pages 162 - 164.*Students to draw out the Tree Diagram for this example as homework.

The above example illustrated the application of Bayes’ Theorem to only two mutually exclusive and collectively exhaustive events, A1 and A2. If there are n such events, A1, A2, …, An, Bayes’ Theorem becomes:

An)|P(An).P(B...A2)|P(A2).P(B A1)|P(A1).P(B

Ai)|P(Ai).P(BB)|P(Ai

Bayes’ Theorem

The multiplication formula indicates that if there are m ways of doing one thing and n ways of doing another thing, there are m x n ways of doing both.Multiplication Formula:Total number of arrangements = (m)(n)Example: A wife told her husband that she needed 10 sets of clothing, so that she would be able to ‘survive’ a two week cycle, without repeating the same set of clothes. Her husband disagreed and said that the same could be achieved with the following: 5 shirts and 2 pants/skirts 2 shirts and 5 pants/skirts- Do you agree?

Multiplication

Permutation Formula:

Where:n is the total number of objects.r is the number of objects selected.

PermutationPermutation: Any arrangement of r objects selected from a single group of n possible objects. The order of arrangement is important in permutations.

r)!-(n

n!Prn

During a S-League match, Coach A had to select 3 players as ‘defender’, out of 11 players. How many ways can the 11 players be arranged in the 3 ‘defender’ positions?

Permutation

990)!311(

!11311

P

CombinationCombinations: The number of ways to choose r objects from a group of n objects without regard to order.

r)!-(nr!

n!Crn

Combination Formula:

Where:n is the total number of objects.r is the number of objects selected.

There are 18 players on the Foxtrot soccer team. Coach A must pick 11 players among the 18 players to comprise the starting team. How many different groups are possible?

Combination

31824)!1118(!11

!181118

C

QUESTIONS ???