What is Probability? (again)

Size: px

Start display at page:

Download "What is Probability? (again)"

Grant Blaze Chambers
5 years ago
Views:

1 INRODUCTION TO ROBBILITY Basic Concepts and Definitions n experient is any process that generates well-defined outcoes. Experient: Record an age Experient: Toss a die Experient: Record an opinion yes, no Experient: Toss two coins The saple space for an experient is the set of all experiental outcoes or siple events. What is robability? In previous lectures we used graphs and nuerical easures to describe data sets which were usually saples. We easured how often using s n gets larger, Relative frequency f/n aple opulation nd How often Relative frequency robability Basic Concepts siple event is the outcoe that is observed on a single repetition of the experient. The basic eleent to which probability is applied. One and only one siple event can occur when the experient is perfored. siple event is denoted by E with a subscript. 4 Basic Concepts Each siple event will be assigned a probability, easuring how often it occurs. The set of all siple events of an experient is called the saple space,. What is robability? again robability is a nuerical easure of the likelihood that an event will occur. robability values are always assigned on a scale fro 0 to 1. probability near 0 indicates an event is very unlikely to occur. probability near 1 indicates an event is alost certain to occur. probability of 0.5 indicates the occurrence of the event is just as likely as it is unlikely. 5 6

2 robability as a Nuerical Measure of the Likelihood of Occurrence robability: Direction of increasing Likelihood of Occurrence The occurrence of the event is just as likely as it is unlikely. ssigning robabilities Classical Method ssigning probabilities based on the assuption of equally likely outcoes. Relative Frequency Method ssigning probabilities based on experiental or historical data. ubjective Method ssigning probabilities based on the assignor s judgent Classical Method If an experient has n possible outcoes, using this ethod one would assign a probability of 1/n to each outcoe. Exaple: Experient: Rolling a die aple pace: {E 1, E, E 3,E 4,E 5,E 6 } robabilities: Each siple event has a 1/6 chance of occurring. Thus we ay generalize: If the siple events in an experient are equally likely, one ay find n nuber of siple events in N total nuber of siple events 9 Relative Frequency Method Exaple: The following table suarizes data on daily rentals of floor polishers for the last 40 days. Nuber of Nuber olishers Rented of Days robability / / etc The probability assignents are given by dividing the nuber-of-days frequencies by the total frequency total nuber of days. 0 ubjective Method When econoic conditions and a copany s circustances change rapidly it ight be inappropriate to assign probabilities based solely on historical data. We can use any data available as well as our experience and intuition, but ultiately a probability value should express our degree of belief that the experiental outcoe will occur. The best probability estiates often are obtained by cobining the estiates fro the classical or relative frequency approach with the subjective estiates. 1 Exaple The die toss: iple events: aple space: 1 E 1 {E 1, E, E 3, E 4, E 5, E 6 } E E 3 6 E 6 E E 4 E 1 E 3 E 5 E 5 E E E 4 E 6

3 Exaple: si six-sided sided fair die toss iple events: E 1, E, E 3,E 4,E 5,E 6, where the subscript denotes the nuber that has occurred on the upper surface of the die. aple space: {E 1, E, E 3,E 4,E 5,E 6 } Event: : an even nuber occurs; B: the nuber is greater than 3; Event can be represented by {E, E 4, E 6 } Event can be represented by B {E 4, E 5, E 6 } Mutually exclusive events: C: the nuber is less than ; D: the nuber is 5 ; How about and C? and D? The robability of an Event The probability of an event easures how often we think will occur; denote. ust be between 0 and 1. If event can never occur, 0. nd on the contrary, if event always occurs when the experient is perfored, 1. The su of the probabilities biliti for all siple events in equals 1. The probability of an event is found by adding the probabilities of all the siple events contained in Basic Concepts n event is a collection of one or ore siple events. 15 The die toss: : an odd nuber B: a nuber > {E 1, E 3, E 5 } B {E 3,E 4,E 5, E 6 } EE 1 E 3 E 5 E E 4 E 6 B 16 Basic Concepts Two events are utually exclusive if, when one event occurs, the other cannot, and vice versa. Experient: Toss a die : observe an odd nuber B: observe a nuber greater than C: observe a 6 D: observe a 3 Mutually Exclusive Not Mutually Exclusive B and C? B and D? 17 The robability of an Event The probability bilit of an event easures how often we think will occur. We write. uppose that an experient is perfored n ties. The relative frequency for an event is Nuber of ties occurs f n n If we let n get infinitely large, f li n n 18 The robability of an Event ust be between 0 and 1. If event can never occur, 0. If event always occurs when the experient is perfored, 1. The su of the probabilities for all siple events in equals 1. The probability bilit of an event is found by adding the probabilities of all the siple events contained in.

4 Finding robabilities robabilities can be found using Estiates t fro epirical i studies Coon sense estiates based on equally likely events. Exaple Toss a fair coin twice. What is the probability of observing at least one head? Exaples: Toss a fair coin. Head 1/ 10% of the U.. population has red hair. elect a person at rando. Red hair.10 1st Coin nd Coin E i E i H HH H T HT H TH T T TT at least 1 head E 1 + E + E / Exaple bowl contains three M&Ms, one red, one blue and one green. child selects two M&Ms at rando. What is the probability that at least one is red? Counting Rules If the siple events in an experient are equally likely, you can calculate 1st M&M nd M&M E i E i RB RG 1/6 1/6 BR 1/6 BG 1/6 GB 1/6 GR 1/6 1 at least 1 red RB + BR+ RG + GR 4/6 /3 n nuber of siple events in N total nuber of siple events You can use counting rules to find n and N. The n Rule 3 If an experient is perfored in two stages, with ways to accoplish the first stage and n ways to accoplish the second stage, then there are n ways to accoplish the experient. This rule is easily extended to k stages, with the nuber of ways equal to n 1 n n 3 n k Exaple: Toss two coins. The total nuber of siple events is: 4 4 Exaples Exaple: Toss three coins. The total nuber of siple events is: 8 Exaple: Toss two dice. The total t nuber of siple events is: Exaple: Two M&Ms are drawn fro a dish containing two red and two blue candies. The total nuber of siple events is: 4 3 1

5 erutations The nuber of ways you can arrange n distinct t objects, taking the r at a tie is n r n! n r! where n! n n 1 n...1 and 0! 1. Exaple: How any 3-digit lock cobinations can we ake fro the nubers 1,, 3, and 4? The order of the choice is iportant! 5 4! ! 4 3 Exaples 6 Exaple: lock consists of five parts and can be assebled in any order. quality control engineer wants to test each order for efficiency of assebly. How any orders are there? The order of the choice is iportant! 5 5! ! Cobinations The nuber of distinct cobinations of n distinct objects that can be fored, taking the r at a tie is n! C n r r! n r! Exaple: Three ebers of a 5-person coittee ust be chosen to for a subcoittee. How any different subcoittees could be fored? 5! !5 3! The order of 5 C3 the choice is 1 not iportant! Exaple box contains six M&Ms, four red and two green. child selects two M&Ms at rando. What is the probability bilit that t exactly one is red? 6! 65! 6 The order of C 15 C1 11!! the choice is!4! 1 not iportant! ways to choose ways to choose M & Ms. 1green M & M. 4 4! C1 4 1!3! ways to choose 1red M & M. 4 8 waysto choose 1 red and 1 green M&M. exactly one red 8/15 Basic Relationships Between Events There are soe basic probability relationships that can be used to copute the probability bilit of an event without t knowledge of all the saple point probabilities. Copleent of event : c consists all saple points that are not in event. Union of events and B: B consists of all saple points belonging to OR BORb both. Intersection of events ND B: B consists of all saple points belonging g to both and B. Mutually Exclusive Events already discussed Event Relations The union of two events, and B, is the event that either or B or both occur when the experient is perfored. We write B B B 9 30

6 Graphical Representation of robability Relationships continued The union of events and B is the event containing all saple points that are in or B or both. The union is denoted by B. The union of and B is illustrated below. aple pace Event Relations The intersection of two events, and B, is the event that both and B occur when the experient is perfored. We write B. B B 31 Event Event B 3 If two events and B are utually exclusive, then B 0. Graphical Representation of robability bilit Relationships concluded d The intersection of events and B is the set of all saple points that are in both and B. The intersection is denoted by B The intersection ti of and B is the area of overlap in the illustration below. Intersection ti aple pace Event Relations The copleent of an event consists of all outcoes of the experient that do not result in event. We write C. C Event Event B Graphical Representation of robability Relationships The copleent of event is defined to be the event consisting of all saple points that are not in. The copleent of is denoted by c. The following diagra below illustrates the concept of a copleent. aple pace Event c Exaple elect a student fro the classroo and record his/her hair color and gender. : student has brown hair B: student is feale C: student is ale What is the relationship between events B and C? 36 C : B C: B C: Mutually exclusive; B C C tudent does not have brown hair tudent is both ale and feale tudent is either ale and feale all students

7 Calculating robabilities for Unions and Copleents There are special rules that will allow you to calculate probabilities for coposite events. The dditive Rule for Unions: For any two events, and B, the probability of their union, B, is 37 B + B B B Exaple: dditive Rule 38 Exaple: uppose that there were 10 students in the classroo, and that they could be classified as follows: : brown hair Brown Not Brown 50/10 Male 0 40 B: feale Feale B 60/10 B + B B 50/ /10-30/10 Check: B 80/10 / /10 pecial Case When two events and B are utually exclusive, B 0 and B + B. : ale with brown hair 0/10 B: feale with brown hair B 30/10 Brown Not Brown Male 0 40 Feale Calculating robabilities for Copleents We know that for any event : C 0 ince either or C ust occur, C 1 so that C + C 1 C and B are utually exclusive, so that B + B 0/ /10 50/10 C Exaple 41 elect a student at rando fro the classroo. Define: : ale 60/10 Brown Not Brown Male 0 40 B: feale Feale and B are copleentary, so that B /10 40/10 Calculating robabilities for Intersections In the previous exaple, we found B directly fro the table. oeties this is ipractical or ipossible. The rule for calculating B depends on the idea of independent and dependent events. 4 Two events, and B, are said to be independent if and only if the probability that event occurs does not change, depending on whether or not event B has occurred.

8 43 Conditional robabilities The probability that occurs, given that event B has occurred is called the conditional probability of f given B and dis defined das B B if B 0 B given 44 Conditional robability The probability of an event given that another event has occurred is called a conditional probability. The conditional probability of given B is denoted by B, where the vertical line stands for given. conditional probability can be coputed by the faous Bayes rule: B B B nd of course the event B ust be such that B 0 Exaple 1 Toss a fair coin twice. Define : head on second toss B: head on first toss Exaple bowl contains five M&Ms, two red and three blue. Randoly select two candies, and define : second candy is red. B: first candy is blue. 45 HH HT TH TT does not change, whether B happens or not B ½ not B ½ and B are independent! 46 B nd red 1 st blue /4 1/ not B nd red 1 st red does change, depending on whether B happens or not and B are dependent! Defining Independence We can redefine independence in ters of conditional probabilities: Two events and B are independent if and only if B or B B Otherwise, they are dependent. d Once you ve decided whether or not two events are independent, you can use the following rule to calculate their intersection. Independence of events Events and B are independent if B. Or, equivalently we ay say Events and B are independent if B B. In other words, the occurrence of event or B has no effect ec on the probability of occurrence ce of event B or

9 Independence of Events continued The ultiplication rule for independent events becoes B B This rule can be used as a test for independence of two events. The Multiplicative Rule for Intersections For any two events, and B,, the probability that both and B occur is B B given that occurred B If the events and B are independent, then the probability that both and B occur is B B Multiplication and ddition Laws for Coputing robabilities The ultiplication law provides a way to copute the probability bilit of an intersection ti of two events. The law is written as: B B B The addition law, on the other hand, provides a way to copute the probability of a union of two events, and writes as follows: B + B - B The addition law for utually exclusive events becoes B + B, since by definition of utually exclusive events B 0 Exaple 1 In a certain population, 10% of the people can be classified as being high risk for a heart attack. Three people are randoly selected fro this population. What is the probability that exactly one of the three are high risk? Define H: high risk N: not high risk exactly one high risk HNN + NHN + NNH HNN + NHN + NNH Exaple uppose we have additional inforation in the previous exaple. We know that only 49% of the population are feale. lso, of the feale patients, 8% are high risk. single person is selected at rando. What is the probability that it is a high risk feale? Define H: high risk F: feale Fro the exaple, F.49 and H F.08. Use the Multiplicative Rule: high risk feale H F FH F The Law of Total robability Let 1,, 3,..., k be utually exclusive and exhaustive events that t is, one and only one ust happen. Then the probability of another event can be written as k k k 53 54

10 The Law of Total robability Let E 1, E, E 3,..., E k be utually exclusive and exhaustive events that is, one and only one ust happen. Then the probability of another event can be written as: E 1 + E + + E k E 1 E 1 + E E + + E k E k The Law of Total robability 1 1 k k k k k k Bayes Rule Let 1,, 3,..., k be utually exclusive and exhaustive events with prior probabilities 1,,, k. If an event occurs, the posterior probability of i, given that occurred is Exaple Fro a previous exaple, we know that 49% of the population are feale. Of the feale patients, 8% are high risk for heart attack, while 1% of the ale patients are high risk. single person is selected at rando and found to be high risk. What is the probability that it is a ale? Define H: high risk F: feale M: ale 57 i i i for i 1,,...k i i 58 We know: M H M.49 M H F M H M + F H F.51 M H F H M.1 Rando Variables quantitative variable x is a rando variable if the value that it assues, corresponding to the outcoe of an experient is a chance or rando event. Rando variables can be discrete or continuous. Exaples: x T score for a randoly selected student x nuber of people in a roo at a randoly selected tie of day x nuber on the upper face of a randoly tossed die 59 robability Distributions for Discrete Rando Variables 60 The probability distribution for a discrete rando variable x resebles the relative frequency distributions we constructed in Chapter 1. It is a graph, table or forula that gives the possible values of x and the probability px associated with each value. We ust have 0 p x 1and p x 1

11 61 Exaple Toss a fair coin three ties and define x nuber of heads. HHH HHT HTH THH HTT THT TTH x TTT 0 x 0 x 1 3/8 x 3/8 x 3 x px 0 1 3/8 3/8 3 robability Histogra for x 6 robability Distributions robability distributions can be used to describe the population, just as we described d saples in Chapter 1. hape: yetric, skewed, ound-shaped Outliers: unusual or unlikely easureents Center and spread: ean and standard d deviation. population ean is called μ and a population standard deviation is called σ. 63 The Mean and tandard Deviation Let x be a discrete rando variable with probability distribution px. Then the ean, variance and standard deviation of x are given as Mean : μ xp x Variance: σ x μ tandard deviation : σ p x σ Exaple Toss a fair coin 3 ties and record x the nuber of heads. x px xpx x-μ px /8 3/ /8 3/8 6/ /8 3 3/ μ xp x 1.5 σ x μ p x σ σ Exaple The probability distribution for x the nuber of heads in tossing 3 fair coins. μ yetric; ound-shaped hape? Outliers? None Center? μ 1.5 pread? σ.688 Key Concepts I. Experients and the aple pace 1. Experients, events, utually exclusive events, siple events. The saple space 3. Venn diagras, tree diagras, probability tables II. robabilities 1. Relative frequency definition of probability. roperties of probabilities a. Each probability lies between 0 and 1. b. u of all siple-event probabilities equals 1. 3., the su of the probabilities for all siple events in

12 Key Concepts III. Counting Rules 1. n Rule; extended n Rule. erutations: n r n! n r! n! 3. Cobinations: C n r r! n r! IV. Event Relations 1. Unions and intersections. Events a. Disjoint or utually exclusive: B 0 b. Copleentary: 1 C Key Concepts B B B 3. Conditional probability: 4. Independent d and dependent d events 5. dditive Rule of robability: B + B B 6. Multiplicative Rule of robability: B B 7. Law of Total robability 8. Bayes Rule Key Concepts V. Discrete Rando Variables and robability Distributions 1. Rando variables, discrete and continuous. roperties of probability distributions 0 p x 1and p x 1 3. Mean or expected value of a discrete rando variable: Mean : μ xp x 4. Variance and standard deviation of a discrete rando variable: Variance: σ x μ p x tandard deviation : σ σ

Key Concepts. Key Concepts. Event Relations. Event Relations

Key Concepts. Key Concepts. Event Relations. Event Relations Probability and Probability Distributions Event Relations S B B Event Relations The intersection of two events, and B, is the event that both and B occur when the experient is perfored. We write B. S Event