Chapter 6 Conditional Probability
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1 Lecture Notes o robability Coditioal robability 6. Suppose RE a radom experimet S sample space C subset of S φ (i.e. (C > 0 A ay evet Give that C must occur, the the probability that A happe is the coditioal probability of A give C. It is deoted by: ( A C ( C Chapter 6 Coditioal robability umber of elemets i A ad C ( A C umber of elemets i C C is called the Reduced Sample Space. A C Example Two fair dice are throw. Give that the sum of the dice is 6, what is the probability that both dice are eve? The sample space is ot relevat to us. Reduced Sample Space RS {sum 6} {(,5, (, 4, (,, (4,, (5, }, (RS 5 A both dice are eve {(,, (, 4, (, 6, (4,, (4, 4, (4,6, (6,, (6,4, (6,6} A C {(, 4, (4, }, (A C ( ( A C A C ( C 5 6. Theorem a ( ( A C A C ( C b roof: a ( A C If S X Y ad X Y φ, the (A (A X(X + (A Y(Y (this is a typical case of theorem of total probability ( A C ( C ( A C ( S ( C ( S ( A C ( C, where S is the sample space. b I the figure, A (A X (A Y ad (A X (A Y φ By axiom of chapter, E A X, F A Y (A (A X + (A Y ( ( A X ( ( A Y A X + ( Y ( X ( Y (A (A X(X + (A Y(Y Example Suppose that 0% of the wome ad 5% of the me at a certai college are members of a badmito club. Moreover, 70% of the studets are male. If a radomly chose studet is a member of the badmito club, what is the probability that the studet is male? Solutio S {college member} X evet of male studets, (X 0.7 Y evet of female studets, (Y 0. X Y φ, X Y S A {badmito club member} (A X 0.05, (A Y 0. (A (A X(X + (A Y(Y (theorem b ( A X ( A X ( X (X A (theorem a ( A ( A (theorem a S S A X X A A C A Y Y 8 Last updated: 8 Jue 05
2 Lecture Notes o robability Chapter 6 Coditioal robability Example Ur A cotais oe red ad two white balls, ur B cotais oe white ad two red balls. A ball is chose at radom from ur A ad trasferred to ur B. Now a ball is chose at radom from ur B, fid the probability that the ball is white? (W? A B Let X evet that the first ball draw from A is red, (X Y evet that the first ball draw from A is white, (Y C evet that the secod ball draw from B is white. By theorem b, (C (C X(X + (C Y(Y 5 Example 4 I a family of childre, give that oe of them is a girl, what is the probability havig aother girl? (Assumig equal probability of boys ad girls. Solutio: S sample space {BG, GB, GG} (aother girl (GG Example 5 eter ad Mary each chose a positive iteger ot exceedig It is kow that eter's umber is divisible by while Mary's umber is divisible by 5. What is the probability that the two umbers are the same? Solutio: Let eter chose the umber x, Mary chose the umber y. The the sample space S {(x, y: x k, y 5m, k, m Z + ad k, m 000} Deote the umber of elemets i the set S by (S. (S If they chose the same umber, the umber may be 5, 0,..., Q , there are 666 favourable outcomes. 666 Required probability Exercise The lock ca be opeed if each row o the lock is tured to the correct umber. There are three rows of umbers ad each row has umbers from 0 to 9. It is kow that at least oe of the three umbers is. If you ca guess oly oce, what is the probability of gettig the correct umber? Aswer Last updated: 8 Jue 05
3 Lecture Notes o robability Example 6 HKCEE 006 aper Q5 Chapter 6 Coditioal robability There are two questios i a test. The probability that David aswers the first questio correctly is 4 ad the probability that David aswers the secod questio correctly is. Give that David aswers at least oe questio correctly i the test, fid the probability that he aswers the secod questio correctly. (He aswers the d correctly he aswers at least oe correctly d ( He aswers the correctly ad at least oe correctly ( at least oe correctly all wrog ( 4 Example 7 Modified from HKCEE 006 aper Q4 There are two classes A ad B takig the same test, each class has 5 studets. The result is as follows: ass Fail Class A 8 7 Class B 0 5 From the 50 studets, studets are radomly selected. Give that exactly of the selected studets pass the test, fid the probability that both of them are i the same class. (both are i the same class exactly studets pass (both are i the same class ad exactly studets pass (exactly studets pass (pass i A, pass i A, fail + (pass (pass, pass, fail or pass, fail, ( i B, pass pass or fail, pass, pass i B, fail Example 8 Modified from HKCEE 007 aper Q5 The followig table shows the results of a survey about the sizes of shirts dressed by 80 studets o a certai school day. Size Studet Small Medium Large Total Boy Girl O that school day, a studet is radomly selected from the 80 studets. Give that the selected studet is a boy, fid the probability that he dresses a shirt of large size. (large shirt boy ( boy ad large shirt ( boy 4 80 Exercise Two cards are draw radomly from ie cards umbered,,, 4, 5, 6, 7, 8 ad 9 respectively oe by oe without replacemet. Give that the sum of the two umbers draw is eve, fid the probability that the secod card draw is odd. 5 Aswer Last updated: 8 Jue 05
4 Lecture Notes o robability Chapter 6 Coditioal robability Example 9 The weights (i kg of a sample of twety F.5 studets are show below (a Show that the mea weight is 66 kg. (b If two of the above data is deleted at radom, fid the probability that the mea weight of the remaiig 8 studets is 65 kg. (c If two of the above data is deleted at radom ad it is kow that oe of them is greater tha 85 kg, fid the probability that the mea weight of the remaiig 8 studets is 65 kg. (b Let the weights of the deleted data be x kg ad y kg x + y 0 66 x + y 50 The favourable outcomes are (60,90, (60, 90, (60, 90, (90, 60, (90, 60, (90, 60, (75, 75, (75, 75 4 robability (c (mea of 8 studets 65 kg deleted oe > 85 kg ( mea 65kg ad deleted oe > 85 kg ( Deleted oe > 85kg ( The deleted two weights are 60kg ad 90 kg ( Deleted oe 90 kg Theorem of total probability Let A, A,..., A be mutually exclusive evets ( is a positive iteger., the (A (A (A A + (A (A A (A (A A roof: A (A A (A A... (A A ad the subsets A A, A A,..., A A are mutually exclusive. By a extesio of axiom i Chapter, (A (A A + (A A (A A ( ( A A A + ( ( A A A A ( A ( A ( A (A (A A + (A (A A (A (A A ( A ( A Bayes Theorem Let A, A,..., A be mutually exclusive evets ad the sample space S A A... A, ( Ak ( A Ak the (A k A roof: (A k A k ( A ( A A k k ( Ak A ( Ak ( Ak A ( Ak ( Ak ( A Ak ( A ( A ( A ( Ak ( A Ak ( A ( A A + ( A ( A A + L ( A ( A A + by the total probability. Last updated: 8 Jue 05
5 Lecture Notes o robability Chapter 6 Coditioal robability Example 0 Suppose that if a perso with SARS disease is give a diagose test the probability that his coditio will be detected is 0.95 ad that if a perso without SARS is give a diagose test the probability that he will be diagosed icorrectly as havig SARS is Suppose, furthermore, that 0. percet of the residets of a certai city have SARS. If oe of these persos (selected at radom, that is, equal probabilities is diagosed as havig SARS o the basis of the diagose test, what is the probability that he actually has SARS? Solutio: (diagose positive SARS 0.95 (diagose positive o SARS 0.00 (SARS 0.00 (SARS diagose ( SARS diagose positive ( diagose positve ( SARS ( diagose positive SARS ( SARS ( diagose positive SARS + ( o SARS ( diagose positive o SARS Example (A aradox Three studets A, B, C are omiated for a scholarship. The three studets have equally soud academic records, so let us assume that each has the same chace of gettig the scholarship. Studet A, kowig the situatio, is eager to have more iformatio. So he asks a professor who already kows the result of the competitio, rofessor, of the other two omiees, please ame oe who will ot get the scholarship?. The professor poders a momet ad replies, studet B. Studet A the goes happily to see his girl fried, sayig, My chace has ow icreased from to, because studet B is out. But what, replied the girlfried, if the professor's aswer is studet C? Well, it's a similar situatio ad my chace would also be icreased to. Come o, laughed the girlfried, if your chace will be whatever the professor's aswer, why did you have to ask? Is t your chace should always be the? Solutio: Questio: Is this a paradox? Assume the professor aswered studet A hoestly. If A gets the scholarship, the the professor may reply B or C with equal chace. If B gets the scholarship, the the professor replies B with o chace. If C gets the scholarship, the the professor must reply B with chace. (A gets the scholarship professor replied studet B ( A ad professor replied "B" ( professor replied "B" ( A ( professor "B" A A professor "B" A + B professor "B" B + C professor "B" ( ( ( ( ( ( C Last updated: 8 Jue 05
6 Lecture Notes o robability Chapter 6 Coditioal robability Example Eric, Fred ad Gary are playig a shootig game together. The probabilities that they hit a flyig target are 0, 5 ad respectively. (a If each of them fires oce, fid the probability that (i all of them hit the target. (ii at least oe of them hits the target. (b If each of them fires twice, fid the probability that (i each of them hits the target at least oce, (ii at least oe of them hits the target oce. (c If each of them fires twice ad it is give that at least oe of them hits the target oce, fid the coditioal probability that Fred hits the target at least oce. Solutio: (a (i (all of them hit the target (ii (at least oe of them hits the target (all of them do t hit the target (b (i (Eric hits the target at least oce (Fred hits the target at least oce (Gary hits the target at least oce (each of them hits the target at least oce (ii 7 (Eric hits the target oce (Fred hits the target oce (Gray hits the target oce (at least oe of them hits the target oce (all of them do't hits the target oce (c (Fred hits the target at least oce at least oe of them hits the target oce (Fred hits the target at least oce ad at least oe of them hits the target oce (at least oe of them hits the target oce Last updated: 8 Jue 05
7 Lecture Notes o robability Chapter 6 Coditioal robability Example Moty Hall roblem Suppose you re o a game show, ad you re give the choice of three doors: Behid oe door is a car; behid the others, goats. You pick a door, say No., ad the host, who kows what s behid the doors, opes aother door, say No., which has a goat. He the says to you, Do you wat to pick door No.? Is it to your advatage to switch your choice? Let C the umber of the door hidig the car, H the umber of the door opeed by the Host. As the car behid each door is equal likely, (C The probability of wiig by switchig the door, give the player choose door o. ((C ad H or (C ad H (C ad H + (C ad H ( C ad H ( C + ( C ad H ( C ( C ( C ( H C + ( H C + If the player switches, the probability of gettig a car is higher. 4 Last updated: 8 Jue 05
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