SChapter 2

UECM2623/UCCM2623 Numerical Methods and Statistics/UECM1693 Mathematics for Physics II

Chapter 2 - 1

Chapter 2: Elementary Probability Theory

2.1 Experiment, Outcomes and Sample Space

Experiment is a process that, when performed, results in one and only one of many observations which are called outcomes of the experiment.

Sample space (denoted by S) is a collection of all outcomes for an experiment.

The elements of a sample space are called sample points.

Example: Experiment Outcomes Sample Space

Toss a coin once Roll a die once Toss a coin twice Take a test Select a student

Head, Tail 1,2,3,4,5,6, HH, HT, TH, TT Pass, Fail Male, Female

S = {Head, Tail} S = {1,2,3,4,5,6} S = {HH, HT, TH, TT} S = {Pass, Fail} S = {Male, Female}

A Venn diagram is a picture that depicts all the possible outcomes for an experiment.

A tree diagram is a picture that represents each outcome by a branch of the tree.

Example 2.1. Draw the Venn and tree diagrams for the experiment of tossing a coin twice.

Solution.

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

Simple event An event that includes one and only one of the (final) outcomes for an experiment, It is usually denoted by Ei .

Compound event Compound event is a collection of more than one outcome for an experiment.

Example 2.2. In a group of people, some are in favor of genetic engineering and others are against it. Two persons are selected at random from this group and asked whether they are in favor of or against genetic engineering. How many distinct outcomes are possible? Draw a Venn diagram and a tree diagram for this experiment. List all the outcomes included in each of the following events and mention whether they are simple or compound events.


Chapter 2 - 2

(a) Both persons are in favor of genetic engineering. (b) At most one person is against genetic engineering. (c) Exactly one person is in favor of genetic engineering.

Solution.

2.2 Counting Sample Points

2.2.1 Multiplicative Rule If an operation can be performed in 1n ways, and if for each of these a second operation can be performed in 2n ways, and for each of the first two a third operation can be performed in 3n ways, and so forth, then the sequence of k operations can be performed in knnn ...21 ways.

Example 2.3. How many sample points are in the sample space when a pair of dice is thrown once?

Solution.

Example 2.4. How many lunches consisting of a soup, sandwich, dessert and a drink are possible if we can select from 4 soups, 3 kinds of sandwiches, 5 desserts and 4 drinks?

Solution.


Chapter 2 - 3

2.2.2 Permutation A permutation is an arrangement of all or part of a set of objects. The number of permutations of n distinct objects is !n .

Example 2.5. How many different arrangements are possible for arranging three letters a, b and c?

Solution.

The number of permutations of n distinct objects taken r at a time is ( )!!rn

nPrn

=

Example 2.6. Two lottery tickets are drawn from 20 for a first and a second prize. Find the number of sample points in the space S.

Solution.

Circular permutations The number of permutations of n distinct objects arranged in a circle is ( )!1n .

The number of distinct permutations of n things of which 1n are of one kind, 2n of a second kind,, kn of a kth kind is

!...!!!

21 knnn

n.

The number of ways of partitioning a set of n objects into r cells with 1n elements in the first cell, 2n elements in the second, and so forth, is

!...!!!

...,,, 2121 rr nnn

n

nnn

n=

where nnnn r =+++ ...21 .

Example 2.7. In how many ways can 7 scientists be assigned to one triple and two double hotel rooms?

Solution.


Chapter 2 - 4

2.2.3 Combination A combination is actually a partition with two cells, the one cell containing the r objects selected and the other cell containing the (n - r) objects that are left.

The number of combinations of n distinct objects taken r at a time is

)!(!!

rnr

n

r

nCr

n

=

=

Example 2.8. From 4 chemists and 3 biologists, find the number of committees that can be formed consisting of 2 chemists and 1 biologist.

Solution.

2.3 Calculating Probability

Probability is a numerical measure of the likelihood that a specific event will occur (denoted by P)

)( iEP = probability that a simple event Ei will occur )(AP = probability that a compound event A will occur

Two Properties of Probability 1. 1)(0 iEP

1)(0 AP

2. =++= 1...)()()( 21 EPEPEP i

2.3.1 Three conceptual approaches to probability i. Classical Probability The classical probability rule is applied to compute the probabilities of events for an experiment

in which all outcomes are equally likely (ie. Each outcome in the sample space has the same probability of occurrence)

If an experiment can result in any one of N different equally likely outcomes, and if exactly n of these outcomes correspond to event A , then the probability of event A is

NnAP =)(


Chapter 2 - 5

Example 2.9. Find the probability of obtaining an even number in one roll of a fair die.

Solution.

ii. Relative frequency Concept of Probability The following probabilities:

- The probability that the next baby born at a hospital is a girl; - The probability that the tossing of an unbalanced coin will result in a head; - The probability that an 80-year-old person will live for at least one more year; Cannot be computed using the classical probability rule because the various outcomes for the corresponding experiments are not equally likely.

To calculate such probabilities, we may perform the experiment again and again to generate data to obtain the relative frequency.

Relative Frequency as an approximation of probability If an experiment is repeated n times and an event A is observed f times, then, according to the relative frequency concept of probability:

n

fAP =)(

Example 2.10. Ten of the 500 randomly selected cars manufactured at a certain auto factory are found to be red in colour. Assuming that the colour of the cars is selected randomly, what is the probability that the next car manufactured at this auto factory is red?

Solution. Let n denote the total number of cars in the sample and f the number of red cars in n .

Law of large numbers

If an experiment is repeated again and again, the probability of an event obtained from the relative frequency approaches the actual or theoretical probability.


Chapter 2 - 6

iii. Subjective Probability Subjective probability is the probability assigned to an event based on subjective judgment,

experience, information and belief.

Examples 1. The probability that Carol, who is taking statistics, will earn an A in this course. 2. The probability that the Dow Jones Industrial Average will be higher at the end of the next

trading day. 3. The probability that Joe will lose the lawsuit he has filed against his landlord.

2.4 Marginal and conditional probabilities

Marginal probability is the probability of a single event without consideration of any other event. It is also called simple probability.

Conditional probability is the probability that an event will occur given that another event has already occurred. If A and B are two events, then the conditional probability of A given B is written as )( BAP It read as the probability of A given that B has already occurred.

Example 2.11. The following is a two way classification of the responses of 100 researchers whether they are in favor of or against genetic engineering.

In Favor (F) Against (A) Total Male (M) 15 45 60

Female (Fe) 4 36 40 Total 19 81 100

Suppose one researcher is selected at random, find the probability that the researcher selected is i. a male. ii. in favor of genetic engineering. iii. against to genetic engineering given that this researcher is a female. iv. a male given that this researcher is in favor of genetic engineering.

Solution.


Chapter 2 - 7

2.5 Mutually Exclusive Events

Events that cannot occur together are said to be mutually exclusive events.

Example 2.12. Consider the following events for one roll of a die: A = an even number is observed; B = an odd number is observed; C = a number less than 5 is observed Are events A and B mutually exclusive? Are events A and C mutually exclusive?

Solution.

2.6 Independent Events

Two events are said to be independent if the occurrence of one does not affect the probability of the occurrence of the other. In other words, A and B are independent events if either )()( APBAP = or )()( BPABP =

If 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.

The two events are dependent if either )()( APBAP or )()( BPABP .

Example 2.13. A box contains a total of 100 CDs that were manufactured on two machines.

Defective (D) Good (G) Total Machine I (A) 9 51 60 Machine II (B) 6 34 40

Total 15 85 100

Are events D and A independent?

Solution.


Chapter 2 - 8

2.7 Complimentary events

The complement of event A, denoted by A is the event that includes all the outcomes for an experiment that are not in A. Therefore, 1)()( =+ APAP .

Example 2.14. Let A be the event that a person has normotensive diastolic blood-pressure ( DBP ) readings ( 90


Chapter 2 - 9

Example 2.15. A box contains 20 DVDs, 4 of which are defective. If 2 DVDs are selected at random (without replacement) from this box, what is the probability that both are defective?

Solution.

2.8.3 Multiplication rule for independent events The probability of the intersection of two independent events A and B is )()()( BPAPBAP =

Example 2.16. The probability that a patient is allergic to penicillin is 0.20. Suppose this drug is administered to three patients. a) Find the probability that all three of them are allergic to it. b) Find the probability that at least one of them is not allergic to it.

Solution.

2.8.4 Joint probability of mutually exclusive events The joint probability of two mutually exclusive events is always zero. If A and B are two mutually exclusive events, then 0)( = BAP .

Example 2.17. Find )( BAP for the events A and B in Example 2.14.

Solution.


Chapter 2 - 10

2.9 Union of events and the addition rule

Union of events Let A and B be two events defined in a sample space. The union of events A and B is the collection of all outcomes that belong either to A or to B or to both A and B and is denoted by A or B (or BA )

Addition rule The probability of the union of two events A and B is )()()()( BAPBPAPBAP += .

Example 2.18. For the following data, what is the probability that a randomly selected person with multiple jobs is a male or single?

Single (A) Married (B) Total Male (M) 1562 2675 4237

Female (F) 1960 1758 3718 Total 3522 4433 7955

Solution.

2.10 Bayes Rule

If the events kBBB ...,,, 21 constitute a partition of the sample space S such that 0)( iBP for ki ...,,2,1= , then for any event A of S,

= =

==k

i

k

iiii BAPBPABPAP

1 1).|()()()(

Bayes Rule If the events kBBB ...,,, 21 constitute a partition of the sample space S, where 0)( iBP for

ki ...,,2,1= , then for any event A in S such that 0)( AP ,

=

= )|()(

)|()()(

)()|(ii

rr

i

r

r BAPBPBAPBP

ABPABP

ABP for kr ...,,2,1=


Chapter 2 - 11

Example 2.19. In a certain assembly plant, three machines 321 ,, BBB make 30%, 45% and 25% respectively, of the products. It is known from past experience that 2%, 3% and 2% of the products made by each machine are defective, respectively. Now, suppose that a finished product is randomly selected. (a) What is the probability that it is defective? (b) If a product were chosen randomly and found to be defective, what is the probability that it was

made by machine 3B ?

Solution.


Chapter 2 - 12

Example 2.20. According to a report, 7.0% of the population has lung disease. Of those having lung disease, 90.0% are smokers; of those not having lung disease, 25.3% are smokers. Determine the probability that a randomly selected smoker has lung disease.

Solution.

SChapter 2

Documents

Transcript of SChapter 2