Objectives CHATER 5 robability and robability Distributions To determine the sample space of experiments To count the elements of sample spaces and events using the fundamental principle of counting, permutation, and combinations To calculate the probability of events using the principles of counting To apply the properties and rules of probability in solving problems involving various types of events Learn the concept of random variables and their probability distributions Apply the normal probability distribution in solving reallife problems Counting Rules Fundamental rinciple of Counting (Multiplicative Rule) Suppose two tasks (activities) can be done in n and m ways, respectively, then there are n m ways of doing the tasks together. Equivalently, suppose each of two trials of an experiment can result in n and m outcomes, respectively, then there are n m possible outcomes of the experiment. Generalization : Suppose each of k tasks can be done in n 1, n,, n k ways, then there are n 1 n n k ways of doing all k tasks together. Counting Rules : How many pairs of a letter and a digit can be made from the letters A, B, and C and the digits 1,, 3, and 4? : How many two-digit numbers of different digits can be formed from the numbers 1,, 3, and 4? : How many outcomes are possible in tossing a coin and a die? : How many outcomes are possible in tossing a coin five times? 1
Counting Rules Factorial Notation The product of all integers less than or equal to n is denoted by n! (read as n factorial ). n 1n n 3... 1 n! n s: 1. 4!=4(3)()(1)=4. 5!=5(4)(3)()(1)=10 3. 6!=6(5)(4)(3)()(1)=70 Counting Rules A permutation is an arrangement of objects where the order of the object matters. : There are 6 different arrangements/permutations of the letters A, B and C. These are: ABC, ACB, BAC, BCA, CAB, CBA. ermutation Rule If r elements are drawn from a set of n distinct or different elements and arranging the r elements in a distinct order, the number of different arrangements is n r n! ( nr )! Counting Rules : In how many ways can 4 people be seated in a room with 9 chairs? : How many 3 digit numbers can be made from the digits 1,,, 9, if each can be used once? How many 7-digit numbers can be made? ermutation Rule If you are arranging N things of which N 1, N,..., N k are alike in the k groups, the number of different arrangements is where N = N 1 + N +... + N k. Counting Rules N, N 1,..., N k N! N! N! N! 1 k : How many different signals can be sent by displaying three flags on a mast if there are six different flags available? : How many different signals can be made by arranging 10 flags in a vertical line if 3 of the flags are red, 3 are blue, are green, and are white?
Counting Rules Combination Rule If you are drawing r elements from a set of n elements without regard to the order of the r elements, the number of different results is n! Cn, r ncr r!n r! : How many committees of 3 members can be formed from 18 people? Counting Rules : In how many ways can a committee of 3 men and women be chosen from 7 men and 5 women? : A class contains 9 boys and 3 girls. i. In how many ways can the teacher choose a committee of 4? ii. How many of them will contain at least one girl? iii. How many of them will contain exactly one girl? What is probability? robability statements are everywhere around us. There is a 65% chance of its raining today. The chance of winning the lotto is 1 in 80 million. There is a 50-50 chance of giving birth to a boy. Randomness and Uncertainty Randomness suggests unpredictability The outcome in tossing a coin cannot be predicted with certainty. At some time or another, everyone will experience uncertainty or doubt. Do you believe you can get a.0 in STAT 1? One might say he is 99% certain of getting a grade of.0 (or even higher) in STAT 1. The concept of probability is used to quantify this measure of doubt. 3
Random experiment Random (Chance) Experiment Any activity or process in which there is uncertainty which of two or more possible outcomes will result. s : tossing a coin rolling a die selecting a card from a deck of playing cards Outcome A particular result of a random experiment. Sample Space (S) Consists of all possible outcomes of an experiment. It is the collection of all simple events Sample point Sample space A single outcome of a random experiment s : 1. Tossing a coin: S={Head, Tail}. Rolling a die: S={1,,3,4,5,6} 3. Sex of children in a two-child family: S={BB,BG,GB,GG} Tree diagram The elements of a sample space may be presented as a tree diagram. : The sex of the children in a two-child family. Tree diagram : An experiment consists of two trials. The first is tossing a coin and observing a head or a tail; the second is rolling a die and observing the number of dots on the upturned face. S = { H1, H, H3, H4, H5, H6, T1, T, T3, T4, T5, T6 } 4
Events We may be interested in only part of the sample space. For example, we may only be concerned with one girl in a two-child family; that is, the outcomes BG and GB. These two outcomes constitute a subset of the sample space and are called events. Each outcome in a sample space is an event; each outcome is a simple event. Every sample point is a simple event. Events are denoted by uppercase letters Events Compound Event Event formed by the combination of two or more simple events. It is formed either by the union or intersection of two or more simple events. The Union of Events The event A or B consists of all outcomes that in A or in B or in both It is denoted by A B : A event of a girl in a two-child family = {BG,GB} Events Events and the Venn diagram Intersection of Events The event A and B consists of all outcomes that are in both of the events A and B. It is denoted by A B. Union of events A and B is shaded Complement of A is shaded Complement of an Event The complement of an event A consists of outcomes in the sample space S but not in A. It is denoted as A (or A c ) Intersection of events A and B is shaded 5
Events : Consider the experiment of tossing a die. Let A be the event of observing an even number of dots and B be the event of observing at least 5 dots. Find a. A B c. A b. A B robability robability: A number between 0 and 1, inclusive, that indicates the likelihood or chance that an event will occur. Sure (certain) event: Event with probability equal to 1. Impossible event: Event with probability equal to 0. Solution: S = {1,, 3, 4, 5, 6} A = {, 4, 6} B = {5, 6} A B = {, 4, 5, 6} A B = {6} A = {1, 3, 5} NOTES: 1. We use to denote probability.. The probability that an event, denoted by A, will occur is symbolized by (A). 3. 0 (A) 1 4. (A i ) = 1, for all A i in S 5. (A)=1-(A c ) Classical probability Relative frequency probability if we can assume that all the simple events in a sample space have the same chance of occurrence, then we can measure the probability of an event as a proportion relative to the number of points in the sample space equivalently, this is equal to the number of simple events in A divided by the number of outcomes in S. A no.of simple events in A n no.of simple events in S n A S : The probability of observing at least one head when two coins are tossed is ¾. probability of an event is the relative frequency at which the event occurs when the random experiment is repeated a large number of times A no.of times A is observed no.of times the experiment is repeated : If 54 heads are recorded when a coin is tossed 100 times, then the probability of a head is estimated to be 0.54. 6
Subjective probability probability of an event is determined based on the knowledge, perception, expertise, and experience : A psychologist believes that men are more likely to be successful entrepreneurs than women. Consider the die tossing experiment. Let A be the event of tossing an even number and B be the event of tossing a number at least 5. Compute the following. a. (A) c. (B ' ) b. (B) Solution: S = {1,, 3, 4, 5, 6} A = {, 4, 6} B = {5, 6} 3 6 A 0. 5 6 B 0. 33 B' 1 0.33 0. 67 Three students are randomly selected and asked of their opinion (Yes or No ) on the RH bill. Construct the sample space. a. What is the probability that at least two students favor the bill? b. What is the probability that all three students do not favor the bill? Solution: Let A=event that at least two students favor the bill. B=event that all three students do not favor the bill S = { NNN, NNY, NYN, NYY, YNN, YNY, YYN, YYY} (A) = (B) = A group of tourists is composed of 3 British, 4 Thais, and Americans. a. How many ways can a tourist guide select 4 persons at random for a trip to Corregidor? b. What is the probability that of the 4 persons, are British and are Thais? c. What is the chance that 1 British,, Thais, and 1 American can go with the trip to Corregidor? ANSWERS: 7
A fair coin is tossed 5 times, and a head (H) or a tail (T) is recorded each time. What is the probability of A = {exactly one head in 5 tosses}, and B = {exactly 5 heads}? Solution: The outcomes consist of a sequence of H s and T s. One outcome may be: HHTTH. There are 3 possible outcomes, all equally likely. A = {HTTTT, THTTT, TTHTT, TTTHT, TTTTH} n( A) ( A) 5 n( S) 3 B = {HHHHH} n( B) ( B) 1 n( S) 3 A quiz consists of five multiple-choice questions. Each question has four choices of which one is correct. Suppose a student answers the quiz by simply guessing. a. How many ways are there to answer the five questions? b. What is the probability of getting all five questions right? c. What is the probability of getting exactly 4 questions right and 1 wrong? d. What is the probability of getting at least 4 questions right? ANSWERS: On the way to work Bob s personal judgment is that he is four times more likely to get caught in a traffic jam (T) than have an easy commute (C). What values should be assigned to (T) and (C)? Solution: 4 T C T C 1 4 C 5 C C C 1 1 1 5 (1) () 1 ( T ) 4 5 4 5 A consumer group studied the service provided by fast-food restaurants in a given community. One of the things they looked at was the relationship between service and whether the server had a high school diploma or not. The information is summarized below. If one server is selected at random, what is the probability that 1. he gives good service?. he has no high school diploma? 8
Addition Rule of robability The probability of the union of events A and B is the sum of the probability of A and the probability of B minus the probability of the intersection of events A and B. (A B) = (A) + (B) - (A B) (A B) = (A) + (B), if A and B are mutually exclusive Mutually Exclusive Events Two events A and B are mutually exclusive if they cannot occur at the same time in a single trial of the experiment. The intersection of mutually exclusive events is the empty (A B= Ø). All employees at a certain company are classified as only one of the following: manager (A), service (B), sales (C), or staff (D). It is known that (A) = 0.15, (B) = 0.40, (C) = 0.5, and (D) = 0.0. What is the probability that a randomly selected employee a. is not the manager? b. is the manager and a service employee? c. is a service or sales employee? d. is not a staff employee? A consumer is selected at random. The probability the consumer has tried a snack food (F) is.5, tried a new soft drink (D) is.6, and tried both the snack food and the soft drink is.. Find the probability that the customer 1. has tried the snack food or the new soft drink.. has not tried the snack food. 3. has tried neither the snack food nor the new soft drink. 4. has tried the soft drink only. The following table summarizes visitors to a local amusement park. All-Day Half-Day ass ass Total Male 100 800 000 Female 900 700 1600 Total 100 1500 3600 One visitor from this group is selected at random. What is the probability that the visitor 1. purchased an all-day pass.. is a male and purchased a half-day pass. 3. is a female or purchased an all-day pass. 9
A manufacturer is testing the production of a new product on two assembly lines. A random sample of parts is selected and each part is inspected for defects. The results are summarized in the table below. Good (G) Defective (D) Total Line 1 (1) 70 40 110 Line () 80 5 105 Total 150 65 15 Suppose a part is selected at random. 1. Find the probability the part is defective.. Find the probability the part is produced on Line 1. 3. Find the probability the part is good or produced on Line. Conditional probability Sometimes two events are related in such a way that the probability of one depends upon whether the second event has occurred. artial information may be relevant to the probability assignment. Given B has occurred, the relevant sample space is no longer S, but B (reduced sample space). Conditional robability The symbol (A B) represents the probability that A will occur given B has occurred. This is called conditional probability. (A B) (A B) (B) Consider the experiment in which a single fair die is rolled: S = {1,, 3, 4, 5, 6 }. 1. What is the probability of a 1 if odd number of dots is observed?. What is the probability of a 1 if even number of dots is observed? 3. What is the probability of an even number if a 4 is observed? From past experience, an instructor estimates that the probability that a student will cheat on an exam is 0.05. The probability that a student cheats and is caught is 0.01. What is the probability that a student will be caught given that the student is cheating? 10
Independent events Two events A and B are independent if the occurrence of one does not affect the probability assigned to the occurrence of the other. Two events are independent if the occurrence of one does not alter the probability of the other Two events A and B are independent events if (A B) = (A) or (B A) = (B) Consider the random experiment of tossing a coin twice. a. What is the probability of observing a tail on the second toss? b. What is the probability of observing tail on the second toss given a tail was observed on the first toss? c. Are the events observing a tail on the first toss and observing a tail on the second toss independent? SOLUTION: If events A and B are not independent, then the events are said to be dependent. Multiplication rule of probability Let A and B be two events defined in sample space S. Then or ( A B) (B) ( A B) (A) If A and B are independent then, (A B) (B A) If A and B are independent, the occurrence of B does not affect the occurrence of A. If A and B are independent, then so are: A A c c Remarks A andb c andb andb c ( A B) (A) (B) 11
In a sample of 100 residents, each person was asked if he or she favored building a new town playground. The responses are summarized in the table below. Age Favor (F) Oppose Total Less than 30 (Y) 50 50 300 30 to 50 (M) 600 75 675 More than 50 (O) 100 15 5 Total 950 50 100 If one resident is selected at random, what is the probability that the resident will: 1. favor the new playground?. favor the playground if the person selected is less than 30? 3. favor the playground if the person selected is more than 50? 4. Are the events F and M independent? Suppose the event A is Allen gets a cold this winter, B is Bob gets a cold this winter, and C is Chris gets a cold this winter. (A) = 0.15, (B) = 0.5, (C) = 0.3, and all three events are independent. Find the probability that 1. All three get colds this winter.. Allen and Bob get a cold but Chris does not. 3. None of the three gets a cold this winter. Remarks 1. Independence and mutually exclusive are two very different concepts. a. Mutually exclusive says the two events cannot occur together, that is, they have no intersection. b. Independence says each event does not affect the other event s probability.. Events cannot be both mutually exclusive and independent. a. If two events are independent, then they are not mutually exclusive. b. If two events are mutually exclusive, then they are not independent. 3. Many probability problems can be represented by tree diagrams. 4. Using the tree diagram, the addition and multiplication rules are easy to apply. Random Variables Random variable is a rule that assigns one (and only one) numerical value to each simple event of an experiment S Outcomes Random Variable 1 0 1 1
Random Variables Three randomly selected potential voters are asked whether they are in favor of a lady president. Each response is recorded as Yes (Y) or No (N). One random variable of interest is the number of voters in favor of a lady president from among the three. Let this random variable be X. S = { NNN, NNY, NYN, YNN, NYY, YNY, YYN, YYY} X=0 X=1 X=3 X= Random Variables Suppose two dice are rolled and the number of dots on the side that lands up on both dice are recorded. The sample space consists of 36 equally-likely outcomes and is shown below. S={(1,1), (1,),, (6,6)} Suppose we are interested on the sum of the dots on the up face on both dice and call this random variable as Y. Then, Y takes values in the set {, 3,, 1} Random Variables Random variables that can assume a finite or countable number of values are called discrete. The random variables X (no. of YES s ) and Y (sum of dots ) in the previous two examples are discrete random variables. Random variables that can assume values corresponding to any of the points contained in one or more intervals on a line are called continuous. The length of time to completely answer all questions in an examination is continuous random variable. robability Distribution of a Discrete Random Variable robability distribution of a discrete random variable is a graph, table, or formula that specifies the probability associated with each possible value of the random variable The probability of each value of the discrete random variable must be greater than or equal to 0: p(x) 0. The sum of the probabilities of all possible values of the discrete random variable must always be 1. 13
Let X - number of heads in a toss of two fair coins. S = { HH, HT, TH, TT } X= {0, 1, } (X=0)=1/4 ; (X=1)=/4; (X=)=1/4 Three randomly selected potential voters are asked whether they are in favor of a lady president. Each response is recorded as Yes (Y) or No (N). One random variable of interest is the number of voters in favor of a lady president from among the three. Let this random variable be X. Construct the probability distribution table of X. X 0 1 (X=x) 1/4 1/ 1/4 x x x X x 0. 5 0. 5,x 0, 1, robability Distribution of a Continuous Random Variable The probability that a continuous random variable takes on an exact value is zero. That is, for a continuous random variable X, (X=x)=0. The probability distribution is specified by the function f(x) with which probability statements are made The function f(x) is referred to as the probability density function of X. This is usually presented by a graph or a formula ( a x b ) a b The total area under the curve is equal to 1. The probability that the random variable X will take on a value between two quantities a and b is given by the area under the curve bounded by the lines X=a and X=b. (X=c)=0 robability Distribution of a Continuous Random Variable f(x) x 14
The Normal Distribution The normal probability distribution is the most important distribution in all of statistics. Many continuous random variables have normal or approximately normal distributions. Normal distribution is perfectly symmetric about its mean μ Its spread is determined by the value of its standard deviation σ Normal probability density function: 1 f ( x) e ( x ) The Normal Random Variable A continuous random variable X is said to be a normal random variable if it follows a normal probability distribution specified by 1 x 1 f ( x) e, x We write X~N(μ,σ ) f(x) roperties of a normal curve 1. It is bell-shaped and unimodal.. It is symmetric at X=μ. 3. It is asymptotic to the X-axis. 4. The total area under the curve is 1. 5. It has a distribution with 68% of the observations fall within the interval [μ σ, μ+σ] 95% of the observations fall within [μ σ, μ+σ] 99.7% of the observations fall within [μ 3σ, μ+3σ] The Standard Normal Distribution It is a normal distribution with mean 0 and variance 1. The random variable which follows a standard normal distribution is referred to as the standard normal variate, denoted by Z. We write Z~N(0,1) Z X 1 1 z 1 f (z) e, z 0 15
The Z Table This table summarizes the cumulative probability distribution for Z, that is, ( Z z ). Table 1. Areas Under the Normal Curve z 0 0.01 0.0 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.5000 0.5040 0.5080 0.510 0.5160 0.5199 0.539 0.579 0.5319 0.5359 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0. 0.5793 0.583 0.5871 0.591 0.5948 0.5987 0.606 0.6064 0.6103 0.6141 0.3 0.6179 0.617 0.655 0.693 0.6331 0.6368 0.6406 0.6443 0.648 0.6517 0.4 0.6554 0.6591 0.668 0.6664 0.6700 0.6736 0.677 0.6808 0.6844 0.6879... 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.861 1.1 0.8643 0.8665 0.8686 0.8708 0.879 0.8749 0.8770 0.8790 0.8810 0.8830 1. 0.8849 0.8869 0.8888 0.8907 0.895 0.8944 0.896 0.8980 0.8997 0.9015 1.3 0.903 0.9049 0.9066 0.908 0.9099 0.9115 0.9131 0.9147 0.916 0.9177 1.4 0.919 0.907 0.9 0.936 0.951 0.965 0.978 0.99 0.9306 0.9319... 3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998 3.5 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 Rules in computing probabilities of normal random variables (Z=a) = 0, hence (Z a) = (Z<a), a>0 (Z a) can be directly obtained from the Z-table (Z>a) = 1 (Z a) (Z> a) = (Z<a) (Z< a) = (Z>a) (a Z b) = (Z b) (Z a) ( a Z a) = (Z a) 1 s on computing probabilities using the Z-table 1. (Z<1.96). (Z>.58) 3. (Z 1.64) 4. ( 1.96 Z 1.96) 5. (Z>.58) 6. (1.96<Z<.58) The average rainfall, recorded to the nearest hundredth of an inch, in a certain municipality, for the month of March was 3.63 inches. Assuming that rainfall in March is normally distributed with a standard deviation of 1.03 inches, find the probability that next March the municipality receives: a. less than 0.7 inch of rain; b. more than inches but not over 3 inches; c. at least 5.3 inches 7. What is the value of a if (Z< a )=0.490? 8. What is the value of a if (Z>a )=0.08? 16
A large group of students took a test in hysics and the final grades have a mean of 70 and a standard deviation of 10. Suppose the passing grade is 60 and we can approximate the distribution of these grades by a normal distribution, what percent of the students a. scored higher than 80? b. should pass the test? c. should fail the test? d. What grade should a student get to be in the top 10% of his class? A radar unit is used to measure speeds of cars on a motorway. The speeds are normally distributed with a mean of 90 kph and a standard deviation of 10 kph. What is the probability that a car picked at random is travelling at more than 100 kph? For a certain type of notebook computers, the length of time between charges of the battery is normally distributed with a mean of 50 hours and a standard deviation of 15 hours. John owns one of such computers and wants to know the probability that the length of time will be between 50 and 70 hours? 17