chapter 4 student

Probability and Statistics I

Chapter 4 Probability

4.1 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 SpaceToss a coin onceRoll a die oncePlay lotteryTake a testSelect a student

Head, Tail1,2,3,4,5,6,Win, LosePass, FailMale, Female

S = { Head, Tail }S = {1,2,3,4,5,6}S = { Win, Lose }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 4.1Draw the Venn and tree diagrams for the experiment of tossing a coin twice.Solution

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

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

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

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Example4.2

Experiment Possible events (a) Roll a die once Event A = roll a 5 =

Event B = roll an even number =(b) Toss two coins Event C = at least one head =

Event D = exactly one head =(c) Demand for a new product Event E = demand is more than 8 =

Example 4.3In 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.(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

Axioms of ProbabilitySuppose S is a sample space associated with an experiment. To every event A in S (A is a subset of S), we assign a number P(A), called the probability of A, so that the following axioms hold :Axiom 1 :Axiom 2 :Axiom 3 : If form a sequence of pairwise mutually exclusive events in S

(that is if i j) then .

4.2 Counting Sample Points

4.2.1 Multiplicative RuleIf an operation can be performed in n1 ways, and if for each of these a second operation can be performed in n2 ways, and for each of the first two a third operation can be performed in n3 ways, and so forth, then the sequence of k operations can be performed in ways.

Example 4.5How many sample points are in the sample space when a pair of dice is thrown once?Solution

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Example 4.6How 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

4.2.2 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!

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

Example 4.7Two 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 permutationsThe number of permutations of n distinct objects arranged in a circle is .

The number of distinct permutations of n things of which n1 are of one kind, n2 of a

second kind, …, nk of a kth kind is .

The number of ways of partitioning a set of n objects into r cells with n1 elements in the

first cell, n2 elements in the second, and so forth, is

where .

Example 4.8In how many ways can 7 scientists be assigned to one triple and two double hotel rooms?Solution

Example 4.9(a) A student is asked to rank five football teams, A, B, C, D, and E in order of his

preference. How many possible ways are there of ordering them 1st, 2nd, 3rd, 4th and 5th?

(b) Find the number of permutations of the letters of the word ‘STATISTICS’.Chapter 4 - 3


(c) How many three digit numbers can be made from the integers 2, 3, 4, 5, and 6 if(i) each integer is used only once;(ii) there is no restriction on the number of times each integer can be used;(iii) the first digit must be 5 and repetition is not allowed?(iv) the first digit must be 5 and repetition is allowed?

Solution

4.2.3 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

Example 4.10From 4 chemists and 3 biologists, find the number of committees that can be formed consisting of 2 chemists and 1 biologist.Solution

Example 4.11(a) In how many ways can five boys be chosen from a class of twenty boys if the

class captain has to be included?(b) A basket of fruits contains a large number of apples, pears, oranges and

bananas. How many different groups of three fruits can be chosen if(i) all of them are of different variety(ii) two of them are of the same variety(iii) all of them are of the same variety?

Solution(a) Number of ways of selecting the captain =

Number of ways of selecting the other 4 boys = Total number of ways of selecting 5 boys with the captain included =

(b) (i) There are 4 different varieties of fruits.If 3 fruits are selected and are of different variety, then 3 varieties are selected.Number of ways of selecting 3 varieties =

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One fruit is taken from each variety. Number of different groups of 3 fruits chosen =

(ii) If 3 fruits are selected and two are of same variety, then 2 varieties are selected.Number of ways of selecting 2 varieties = From the 2 selected varieties, number of ways of selecting 1 variety where two fruits are taken =

Number of different groups of 3 fruits chosen =

(iii)If 3 fruits are selected and are of the same variety, then 1 variety is selected.Number of ways of selecting 1 variety = All three fruits are taken from the selected variety. Number of different groups of 3 fruits chosen =

4.3 Calculating Probability

Probability is a numerical measure of the likelihood that a specific event will occur.

= probability that a simple event Ei will occur = probability that a compound event A will occur

Two properties of probability1. 0 1

0 12.

4.3.1 Three conceptual approaches to probability

I. Classical ProbabilityThe 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 .

Example 4.12A fair die is thrown. Let A be the event “the number is odd” and B be the event “the number is greater than 4”.(a) State the sample space.

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(b) Find P(A) and P(B).Solution

II. Relative frequency Concept of ProbabilityThe 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 probabilityIf an experiment is repeated n times and an event A is observed f times, then, according to the relative frequency concept of probability:

P(A)

Example 4.13A survey of 350 families gave the following data on the number of children under 16 years old in each family:

Number of children under 16 years old Frequency0 1851 512 90

3 or more 24Total 350

Find the probability of the following events :(a) A, a household selected has no children under 16 years old;(b) B, a household selected has at least one child under 16 years old.Solution

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

III. Subjective ProbabilityChapter 4 - 6


Subjective probability is the probability assigned to an event based on subjective judgment, experience, information and belief.

Examples1. 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.

Probability and Combination AnalysisIn most of the probability problems, we need to determine the number of possible outcomes in the sample space as well as the event. This is usually involves permutation and combination.

Example 4.14Five cards are drawn from a pack of 52 well-shuffled cards. Find the probability that(a) 4 are aces(b) 4 are aces and 1 is king(c) 3 are tens and 2 are jacks(d) a 9, 10, jack, queen, king are obtained in any order

Solution(a) P(4 are aces) =

(b) P(4 aces and 1 king) =

(c) P(3 are tens and 2 are jacks) =

(d) P(9, 10, jack, queen, king) =

4.4 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 .It read as “the probability of A given that B has already occurred”.

Example 4.11The following is a two way classification of the responses of 100 researchers whether they are in favor of or against genetic engineering.

In Favor Against TotalChapter 4 - 7


Male 15 45 60Female 4 36 40Total 19 81 100

Suppose one researcher is selected at random, find the probability that the researcher selected is(a) a male.(b) in favor of genetic engineering.(c) against to genetic engineering given that this researcher is a female. (d) a male given that this researcher is in favor of genetic engineering.

Solution

4.5 Mutually Exclusive Events

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

Example 4.12Consider the following events for one roll of a die :

A = an even number is observedB = an odd number is observedC = a number less than 5 is observed

Are events A and B mutually exclusive? Are events A and C mutually exclusive?

Solution

4.6 Independent EventsTwo 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 or

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 or .

Example 4.13A box contains a total of 100 CDs that were manufactured on two machines.

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

Total 15 85 100Chapter 4 - 8


Are events D and A independent?

Solution

Two important observations about mutually exclusive, independent, and dependent events.

1. Two events are either mutually exclusive or independent.a. Mutually exclusive events are always dependent.b. Independent events are never mutually exclusive.

2. Dependent events may or may not be mutually exclusive.

4.7 Complimentary eventsThe complement of event A, denoted by is the event that includes all the outcomes for an experiment that are not in A. Therefore, .

Example 4.14In a group of 2000 taxpayers, 400 have been audited by the IRS at least once. If one taxpayer is randomly selected from this group, what are the two complementary events and their respective probabilities?Solution

4.8 Intersection of events and the multiplicative rule

Intersection of eventsLet A and B be two events defined in a sample space. The intersection of A and B represents the collection of all outcomes that are common to both A and B and is denoted by “A and B” (or )

Joint ProbabilityThe probability of the intersection of two events is called their joint probability and written as

Multiplicative Law of ProbabilityThe probability of the intersection of two events A and B is

The probability of the intersection of two independent events A and B is

Conditional ProbabilityChapter 4 - 9


If A and B are two events, then and given that

and .

Example 4.15A 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

Example 4.16The 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

Joint probability of mutually exclusive eventsThe joint probability of two mutually exclusive events is always zero. If A and B are two mutually exclusive events, then .

4.9 Union of events and the addition rule

Union of eventsLet 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 .

The Additive Law of ProbabilityThe probability of the union of two events A and B is If A and B are two mutually exclusive events, then and

Example 4.17Let A be the event that a person has normotensive diastolic blood-pressure ( ) readings ( ) and let B be the event that a person has borderline readings ( ). Suppose , .(a) Let C be the event that a person has , find .(b) Let D be the event that a person has , find .(c) Two persons are randomly selected, find the probability that at least one of them

has .Solution

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Example 4.18For the following data, what is the probability that a randomly selected person with multiple jobs is a male or single?

Single (A) Married (B) TotalMale (M) 1562 2675 4237

Female (F) 1960 1758 3718Total 3522 4433 7955

Solution

4.10 Bayes’ RuleFor some positive integer k, let the sets be such that 1.2. if Then the collection of sets is said to be a partition of S.

If the events constitute a partition of the sample space S such that for , then for any event A of S,

Total Probability

Bayes’ RuleIf the events constitute a partition of the sample space S, where for , then for any event A in S such that ,

for Posterior Probability

Example 4.19According 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

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Example 4.20In a certain assembly plant, three machines 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 (i) it was made by machine ?(ii) it was not made by machine ?

Solution

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