chapter 2brahms.emu.edu.tr/zorlu/chapter5m211.doc · web viewchapter 5 probability and some...

Math 211 Introduction to Statistics

Chapter 5 Probability and Some Probability Distributions

Key words: Sample space, sample point, tree diagram, events, complement, union and intersection of an event, mutually exclusive events; Counting techniques: multiplication rule, permutation, combination, probability of an event; Additive rules; Conditional probability, independent events, multiplicative rules, Bayes’ rule.

SAMPLE SPACE

Sample Space. The set of all possible outcomes of a statistical experiment is called a Sample space. It is represented by the symbol S.

Sample Point. Each outcome in a sample space is called an element or a sample point of the sample space.

Example 1. (a) Consider the experiment of tossing a coin. The sample space S of possible outcomes may be written as

S= .

(b) Consider the experiment of flipping a die. Then the elements of the sample space S is listed as

S= .(c) Now consider the experiment of tossing a die and then a coin once. The resultantsample space can be obtained using TREE DIAGRAM. That is,

Therefore the sample sample S is S=.

Sonuc Zorlu Lecture Notes1


Event. An event is a subset of a sample space.

For example is an event defined on S. One can define events on S.Empty set , is an impossible event and S is a sure event. Any subset of S is represented by capital letters such as A, B, C…

The Complement of an Event. The complement of an event A with respect to S is the subset of all elements of S that are not in A. The complement of A is denoted by the symbol or .

The Intersection of Events. The intersection of two events A and B, denoted by the symbol is the event containing all elements that are common to A and B.

Mutually Exclusive Events. Two events A and B are mutually exclusive or disjoint if , that is if A and B have no common elements in common.

The Union of Events. The union of two events A and B, denoted by the symbol is the event containing all elements that belong to A or B or both.

Important Notes. The following results may easily be verified by means of Venn diagrams.(1) (2) (3) (4) (5) (6) (7)

(8) 1st De Morgan Rule

(9) 2nd De Morgan Rule

Example 2. If and , ,

and , list the elements of the sets corresponding to the following events: (a) (b) ( and are mutually exclusive events)(c) (d)

(e)

(f)

COUNTING SAMPLE POINTS



Multiplication Rule. If an operation can be performed in ways, and if for each of these a second operation can be performed in ways, then the two operations can be performed together in ways.

Example 1. How many breakfasts consisting of a drink and a sandwich are possible if we can select from 3 drinks and 4 kinds of sandwiches?

Solution : and , therefore there are different ways to choose a breakfast.

Example 2. A certain shoe comes in 5 different styles with each style available in 4 distinct colors. If the store wishes to display pairs of these shoes showing all of its various styles and colors, how many different pairs would the store have on display?

Solution : different pairs are available.

Example 3. In how many ways can a true-false test consisting of 9 questions be answered?

Solution : There are different answers for this test.

Example 4. How many even three-digit numbers can be formed fro the digits 1,2,5,6, and 9 if (a) each digit can be used only once?(b) each digit can be repeated?

Solution : (a) Since the number must be even we have only 2 choices for the units position. For each of these there are 4 choices for hundreds position and 3 choices for tens positions. Therefore we can form a total of even three-digit numbers.

(b) Similarly for the units position there are 2 choices, since the numbers can be repeated we have 4 choices for the hundreds and tens positions. Hence, totally there are even three-digit numbers.

Permutation . A permutation is an arrangement of all or part of a set of objects.

The number of permutations of distinct objects is .

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

Example 5 . In how many ways can a Society schedule 3 speakers for 3 different meetings if they are available on any of 5 possible dates?



Solution : The total number of possible schedules is

.

Theorem . The number of permutations of distinct objects arranged in a circle is .

Theorem . The number of distinct permutations of things of which are of one kind, of a second kind,…, of a kind is

Example 6. How many different ways can 3 red, 4 yellow, and 2 blue bulbs be arranged in a string of Christmas tree lights with 9 sockets?

Solution : The total number of distinct arrangements is

.

Example 7. Suppose that 10 employees are to be divided among three jobs with 3 employees going to job I, 4 to job II, and 3 to job III. In how many ways can the job assignment be made?

Solution. =2200

Theorem. The number of combinations of distinct objects taken at a time is

Example 8. From 4 mathematicians and 6 computer scientists, find the number of committees that can be formed consisting of 2 mathematicians and 4 computer scientists.

Solution.

Probability of an Event



Definition. Suppose that an experiment has associated with it a sample space S. A probability is a numerically valued function that assigns a number to every event so that the following axioms hold:

(1) (2) (3) If is a sequence of mutually exclusive events (i.e. for any

), then

Example 1. A coin is tossed twice. What is the probability that at least one head occurs?

The sample space for this experiment is . If the coin is balanced, each of these outcomes would be equally likely to occur. Therefore we assign a probability w to each sample point. Then 4w=1 or w=1/4.If A represents the event of at least one head occurring, then

.

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

Additive Rules

Theorem. If and are two events, then

Corollary. If and are mutually exclusive events, then

Corollary. If are mutually exclusive events, then

Theorem. For three events , and ,

Example 1. What is the probability of getting a total 7 or 11 when a pair of dice are tossed?

Solution. Let ; , therefore



Theorem. If and are complementary events, then

Example 2. In a random experiment it is known that , Calculate .

Exercises

Exercise 1. A pair of dice is tossed. Find the probability of getting(a) a total of 8(b) at most a total of 5.

Exercise 2. Two cards are drawn in succession from a deck without replacement. What is the probability that both cards are greater than 3 and less than 9?

Exercise 3. If 3 books are picked at random from a shelf containing 5 novels, 3 books of poems, and a dictionary, what is the probability that

(a) the dictionary is selected?(b) 2 novels and 1 book of poems are selected?

Exercise 4. If 2 of the 10 employees are female and 8 male, what is the probability that exactly one female gets selected among the three?

Exercise 5. A package of 6 light bulbs contain 2 defective bulbs. If 3 bulbs are selected for use, find the probability that none is defective.

Exercise 6. How many four-digit serial numbers can be formed if no digit is to be repeated within any one number?

Exercise 7. Seven applicants have applied for two jobs. How many ways can the jobs be filled if (a) the first person chosen receives a higher salary than the second?(b) There are no differences between the jobs?

Exercise 8. From a group of 4 men and 5 women, how many committees of size 3 are possible(a) with no restrictions?(b) with 1 man and 2 women?(c) with 2 men and 1 woman if a certain man must be on the committee?

Exercise 9. In how many ways can the letters of the word ‘KRAKATOA’ be arranged?

Random Variables and Probability Distributions



In statistics we deal with random variables- variables whose observed value is determined by chance. Random variables usually fall into one of two categories: discrete or continuous

Random Variable. A random variable (r.v.) is a function that associates a real number with each element in the sample space. Random variables will be denoted by uppercase letters and their observed numerical values by lowercase letters.

Discrete Random Variable. A random variable is discrete if it can assume at most a finite or a countably infinite number of possible values.

Example 1. Two balls are drawn in succession without replacement from an urn containing 4 red and 3 black balls. The possible outcomes and values of the random variable where is the number of red balls are:

Sample SpaceRR 2RB 1BR 1BB 0

Continuous Random Variable. A random variable is continuous if it can assume any value in some interval or intervals of real numbers and the probability that it assumes any specific value is 0.

Discrete Probability Distributions

Definition. The set of ordered pairs is a probability function, probability mass function or probability distribution of the discrete random variable if,

(1)

(2)

(3) .

Example 1. A committee of size 5 is to be selected at random from 3 chemists and 5 mathematicians. Find the probability distribution (p.d.) for the number of chemists on the committee.

Let be the number of chemists on the committee. Then .



;

;

Therefore the probability distribution of is

0 1 2 3

Exercise 1. Among 10 applicants for an open position 6 are female and are males. Suppose 3 applicants are randomly selected from the applicant pool for final interviews. Find the probability distribution for , which is the number of female applicants among the final three.

Exercise 2. Let be a random variable giving the number of heads minus the number of tails in three tosses of a coin.

(a) List the elements of the sample space(b) Assign a value of to each sample points.(c) Find the probability distribution of the random variable assuming that the coin is

biased so that a head is twice as likely to occur as a tail.

Cumulative Distribution. The cumulative distribution of a discrete random variable

with probability distribution is

, for .

Example 2. Find the cumulative distribution of the number of red balls in example 1. Using

, show that .

;



;

.

Hence,

Now,

Continuous Probability Distributions

Definition. (Probability Density Function) The function is a probability density function for the continuous random variable , defined over the set of real numbers , if

(1)

(2)

(3) .

Note that for a continuous random variable ,

.

Cumulative Distribution. The cumulative distribution of a continuous random variable

with density function is

.



Example 1. Suppose that a random variable has a probability density function given by

(a) Find the value of that makes this a probability function.

(b) Find

(c) Find and sketch the graph of this function.

Some Useful Probability Distributions

The observations generated by different statistical experiments have the same general type of behavior. Random variables associated with these experiments can be described by essentially the same probability distribution and therefore can be represented by a single formula. The followings are the probability distributions that will be covered in this chapter:

Binomial Distribution Poisson Distribution and Poisson Process Normal Distributions

The Binomial Distribution

Perhaps the most commonly used discrete probability distribution is the binomial distribution. An experiment which follows a binomial distribution will satisfy the following requirements (think of repeatedly flipping a coin as you read these):

1. The experiment consists of n identical trials, where n is fixed in advance. 2. Each trial has two possible outcomes, S or F, which we denote ``success'' and ``failure''

and code as 1 and 0, respectively. 3. The trials are independent, so the outcome of one trial has no effect on the outcome of

another. 4. The probability of success, is constant from one trial to another.



The random variable X of a binomial distribution counts the number of successes in n trials. The probability that X is a certain value x is given by the formula

where and . Recall that the quantity , ``n choose x,'' above is

We could use the formulas previously given to compute the mean and variance of X. However, for the binomial distribution these will always be equal to

and

NOTE:\ A particularly important example of the use of the binomial distribution is when sampling with replacement (this implies that is constant).

Example 1. Suppose we have 10 balls in a bowl, 3 of the balls are red and 7 of them are blue. Define success S as drawing a red ball. If we sample with replacement, P(S)=0.3 for every trial. Let's say n=20, then and we can figure out any probability we want. For example,

The mean and variance are

Example 2. The probability that a patient recovers from a rare blood disease is 0.4. If 15 people are known to have contracted this disease, what is the probability that

(a) at least 10 survive?(b) from 3 to 7 survive?(c) exactly 5 survive?

Solution. Probability of success= , and the probability of failure= .and no. of surviving patients

(a) 1-0.9662=0.0338

(b)

(c) 0.186

Example 3. A traffic control engineer reports that 75% of the vehicles passing through a checkpoint are from within the state. What is the probability that fewer than 2 of the next 9 vehicles are from out of the state?



Solution. Probability of success= , and the probability of failure= .and no. of vehicles passing through the checkpoint

Example 4. Assuming that 6 in 10 automobile accidents are due mainly to speed violation, (a) find the probability that among 8 automobile accidents 6 will be due mainly to a

speed violation.(b) Find the mean and variance of the number of automobile accidents for 8 automobile

accidents.

Solution. Probability of success= , and the probability of failure= .and no. of automobile accidents

(a) 0.2090

(b) The mean of the number of automobile accidents is .

The variance of the no. of auto. accidents is .

The Poisson Distribution

The Poisson distribution is most commonly used to model the number of random occurrences of some phenomenon in a specified unit of space or time. For example,

The number of phone calls received by a telephone operator in a 10-minute period. The number of flaws in a bolt of fabric. The number of typos per page made by a secretary.

For a Poisson random variable, the probability that X is some value x is given by the formula

where is the average number of occurrences per unit time or region denoted by . For the Poisson distribution,

and .



Example 1. The number of false fire alarms in a suburb of Houston averages 2.1 per day. Assuming that a Poisson distribution is appropriate, the probability that 4 false alarms will occur on a given day is given by

Example 2. During a laboratory experiment the average number of radioactive particles passing through a counter in 1 millisecond is 4. What is the probability that 6 particles enter the counter in a given millisecond?

Example 3. A secretary makes 2 errors per page, on average. What is the probability that on the next page he or she will make

(a) 4 or more errors?(b) no errors?

Example 4. The number of customers arriving per hour at a certain automobile service facility is assumed to follow a Poisson distribution with .

(a) Compute the probability that more than 10 customers will arrive in a 3-hour period.(b) What is the mean number of arrivals during a 4-hour period?

Example 5. A restaurant chef prepares tossed salad containing, on average, 5 vegetables. Find the probability that the salad contains more than 5 vegetables

(a) on a given day(b) on 3 of the next 4 days(c) for the first time in April on April 5.

Poisson Approximation to Binomial Distribution

Theorem. Let be a binomial random variable with probability distribution . When and remains constant,

.

Example 1. The probability that a person will die from a certain respiratory infection is 0.002. Find the probability that fewer than 5 of the next 2000 so infected will die.

number of infected people who will die

Given , ,



Example 2. It is known that 5% of the books bound by a certain bindery have defective bindings. Find the probability that 2 of 100 books bound by this bindery will have defective bindings using

(a) the formula for the binomial distribution(b) the Poisson approximation to the Binomial distribution.

Solution. (a) Given , and , number of defective bindings,

(b) where .

The Normal Distribution

The most important continuous probability distribution in the entire field of statistics is the normal distribution. Normal distributions are a family of distributions that have the same general shape. They are symmetric with scores more concentrated in the middle than in the tails. Normal distributions are sometimes described as bell shaped which are shown below. Notice that they differ in how spread out they are. The area under each curve is the same. The height of a normal distribution can be specified mathematically in terms of two parameters: the mean (μ) and the standard deviation (σ).

Definition. The density function of the normal random variable , with mean and variance , is

where .


http://davidmlane.com/hyperstat/A16252.html




Standard normal distribution

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. Normal distributions can be transformed to standard normal distributions by the formula:

where X is a score from the original normal distribution, μ is the mean of the original normal distribution, and σ is the standard deviation of original normal distribution. The standard normal distribution is sometimes called the z distribution. A z score always reflects the number of standard deviations above or below the mean a particular score is. For instance, if a person scored a 70 on a test with a mean of 50 and a standard deviation of 10, then they scored 2 standard deviations above the mean. Converting the test scores to z scores, an X of 70 would be:

So, a z score of 2 means the original score was 2 standard deviations above the mean. Note that the z distribution will only be a normal distribution if the original distribution (X) is normal.

Note : The following figures give us the areas to the right of some value, to the left of some value and between two values.







Areas under portions of the standard normal distribution are shown to the right. About .68 (.34 + .34) of the distribution is between -1 and 1 while about .96 of the distribution is between -2 and 2.

Example 1. Given a standard normal distribution, find the area under the curve that lies (a) to the right of (b) between

Solution. (a) .

(b)

Example 2. Given a normal distribution with and , find the probability that assumes a value between 45 and 62.

Solution.

Example 3. Given a standard normal distribution, find the value of such that(a)

(b)

(c)

Solution. (a)

(b)

(c)



Example 4. Given the normally distributed random variable with and

(a) Compute

(b) Compute satisfying .

Applications of the normal Distribution

Example 1. A certain machine makes electrical resistors having a mean resistance of 40 ohms and a standard deviation of 2 ohms. Assuming that the resistamce follows a normal distribution and can be measured to any degree of accuracy, what percenatge of resistors wil have a resistance exceeding 43 ohms?

Solution. Let be the normal random variable, given ,

Example 2. The average grade for a exam is 74 and the standard deviation is 7. Assuming that the grades follow a normal distribution, what is the probability that a student will get grade of at least 60?

Normal Approximation to Binomial Distribution

Theorem. If is a binomial random variable with mean and , then the limiting form of distribution of

as

is the standard normal distribution .

Note. We use the normal approximation to binomial distribution whenever is not close to 0 and 1. If both and are greater than or equal to 5, the approximation will be good.

Example 1. The probability that a patient recovers from a blood disease is 0.4. If 100 people are known to have contracted this disease what is the probability that less than 30 survive?

Solution. Let be the number of surviving people from blood disease.

Given , , ,



Example 2. A coin is tossed 400 times, use the normal approximation to binomial to find the probability of obtaining

(a) between 185 and 210 heads inclusive(b) exactly 205 heads(c) less than 176 or more than 227 heads

Example 3. A ablanced die is rolled 180 times. Let be the number of cases when die shows the number 4 on its top face.

(a) Find (b) Find

(c) Use normal approximation to binomial to approximate .

Exercises

Exercise1. If scores are normally distributed with a mean of 30 and a standard deviation of 5, what percent of the scores is: (a) greater than 30? (b) greater than 37? (c) between 28 and 34?

Exercise 2. Assume a normal distribution with a mean of 90 and a standard deviation of 7. What limits would include the middle 65% of the cases?

Exercise 3. If is the standard normal random variable,(a) Calculate

(b) If , find the value of .

Exercise 4. A research scientists reports that mice will live an average of 40 months when their diets are sharply restricted and enriched with vitamins and proteins. Assuming that the lifetimes of such mice are normally distributed with a standard deviation of 6.3 months, find the probability that a given mouse will live

(a) more than 32 months(b) less than 28 months(c) between 37 and 49 months


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