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1 ocepts ad importat formulae o probability Key cocept: *coditioal probability *properties of coditioal probability *Multiplicatio Theorem o Probablity *idepedet evets *Theorem of Total Probablity *Bayes Theorem *Probability Distributio of a radom variable *Mea, Variace ad Stadard deviatio of a radom variable *Beroulli trials ad Biomial Distributio The coditioal probability ; If E ad F are two evets associated with the same sample space of a radom experimet,the coditioal probability of the evet E, give the occurrece of the evet F is give by ) Properties of Probability : Let E ad F be evets of a sample space S of a experimet, the we have (a) P( s )P F (F ) (b) 0 P(E/F), (c) F P(E /F) -P(E/F) (d) P((E F)/G) P(E/G) + P(F/G) P((E F)/G) ) Multiplicatio Theorem o Probability P(E F) P(E) P(F/E), P(E) 0 P(E F) P(F) P(E/F), P(E) 0 3) If E & F are idepedet, the P(E F) P(E) P(F) P(E/F) P(E), P(F) 0 P(F/E) P(F), P(E) 0

2 4) THEOREM OF TOTAL PROBABILITY sthe evet E, E,E3,..E has o-zero probability. Let A be ay evet associated with S, the P(A) P(E) P(A/E) + P(E) P(A/E) +..+ P(E) P(A/E) ) BAYE S THEOREM If E, E,,E are the evets which costitute a partitio of S i.e E,E,,E are pair wise disjoit & E E.. E S & A be ay evet with o-zero probability, the P (Ei/A) P(Ei)P(A/Ei) j P(Ej)P(A/Ej) 6) A radom variable is a real valued f. whose domai is the sample space of radom experimet. 7) The probability distributio of a radom variable X is the system of os. 8) X : x x.. x P(X) : p p p Where, pi>0, i pi,i,,., 9) Let X be a Radom variable whose possible values x,x,..,x occur with probabilities p,p,..,p resp. The mea of X, deoted byμ, is the No. xipi i. The mea of Radom variable X is also called the Expectatio of X, deoted by E(X). 0) Let X be a Radom Variable whose possible values x,x,.,x occur with probabilities p(x),p(x),..,p(x) resp. Let μ E(X) be The mea of X. The variace of X,deoted by Var(X) or σx,is defied as σx Var(X) i(xi μ) p(xi) or σx E(X-μ ) The o-egative o. σ x Var(X) i(xi μ) p(xi), is called the stadard deviatio of the Radom VarialeX. ) Var(X) E(X ) - [E(X)] ) Trials of a radom experimet are called Beroulli trials, if they satisfy the followig coditios. i) There should be a fiite o. of trials. ii) The trial should be idepedet. iii) Each trial has exactly two outcomes: success or failure

3 iv) The probability of success remais the same i each trial. For Biomial distributio B(,p), P(Xx) xq -x p x, x 0,,,., & (q -p Mea p, ad Variece pq Stadard deviatio pq IMPORTANT BOARD QUESTIONS Q. A Isurace ompay isured 000 scooter drivers, 4000 car drivers ad 6000 truck drivers. The probability of a accidet ivolvig a scooter, a car ad a truck is 0.0, 0.03 ad 0. respectively. If a driver meets a accidet, what is the chace that the perso is a scooter driver? What is importace of isurace i everybody s life. Q.. A card from a pack of cards is lost. From the remaiig cards of the pack, two cards are draw ad are foud to be both diamods. Fid the probability of the lost card beig a diamod. Does it better ot to tell ay perso regardig loss of the card while playig? Q.3 A ma is kow to speak truth 3 out of 4 times. He throws a die ad reports that it is a six. Fid the probability that it is actually a six. Write at least oe drawback of tellig lie. Q.4 A factory has two machies A ad B. Past record shows that machie A produced 60% of the items of output ad machie B produced 40% of the items. Further, % of the items produced by machie A ad % produced by machie B were defective. All the items are put ito oe stockpile ad the oe item is chose at radom from this ad is foud to be defective. What is the probability that it was produced by machie B? Q.4. A ur cotais red ad black balls. A ball is draw at radom, its colour is oted ad is retured to the ur. Moreover, additioal balls of the colour draw are put i the ur ad the a ball is draw at radom. What is the probability that the secod ball is red? Q.. A laboratory blood test is 99% effective i detectig a certai disease whe it is i fact, preset. However, the test also yields a false positive result for 0.% of the healthy perso tested (that is, if a healthy perso is tested, the, with probability 0.00, the test will imply he has the disease). If 0. percet of the populatio actually has the disease, what is the probability that a perso has the disease give that his test result is positive? Q.6 O a multiple choice examiatio with three possible aswers (out of which oly oe is correct) for each of the five questios, What is the probability that a cadidate would get four of or more correct aswers just by guessig? (009) Q.7. Fid the probability of throwig at most sixes i 6 throws of a sigle throw a die. Q.8 A experimet succeeds thrice as ofte as it fails. Fid the probability that the ext five trails, there will be at least 3 successes. (04)

4 Q.9 How may times must a fair coi be tossed so that the probability of gettig atleast oe head is more tha 80%? (0) 0. A radom variable has followig probability distributio: X P(X) 0 k k k 3k k k 7k + k Determie: (i) k (ii) P(X < 3) (iii) P(X > 6) (iv) P(0 < X < 3) Q.. Two cards are draw simultaeously (without replacemet) from a well- shuffled pack of cards. Fid the mea ad variace of the umber of red cards. (0) Q..Two umbers are selected at radom without replacemet from first 6 positive itegers,let X deote the largest of the two umbers obtaied. Fid the probability distributio of X also fid the expectatio of X. ANSWERS Q.. Let A be the evet that the isured perso meets with a accidet ad E, E ad E3 are the evets that the perso is a scooter, car ad truck driver respectively. The we have to fid P(E/A). Total umber of isured persos P (E) 000/000 /6; P (E) 4000/000 /3; P (E3) 6000/000 ½ Also P (A/E) 0.0; P (A/E) 0.03 ad P (A/E3) 0. 3marks Hece by Bye s theorem we have P E / A P A / E P E P A / E P E P A / E P E P A / E P E / / / 3 0. / /. Q. let E,E,E3,E4 ad A be the evets defied as follows; Ethe missig card is a diamod Ethe missig card is ot diamod A two draw card are of diamod Now P(E)3/ ¼

5 P(E) ¾ P(A/E) probability of drawig of secod heart cards whe oe diamod card is missig (,). (,) 0 Similarly P(A/E) (3,) 3. (,) 0 P ( E A ) A P(E ). P( ) E P(E ). P ( A E ) + P(E ). P( A E ) No, Whe we are playig ay game it should be played hoestly.

6 Q.3 The evet that six occurs ad S be the evet that six does ot occur. The P(S) Probability that six occurs 6 P(S) Probability that six does ot occur 6 P(E S) Probability that the ma reports that six occurs whe six has actually occurred o the die Probability that the ma speaks the truth 3 4 P(E S) Probability that the ma reports that six occurs whe six has ot actually occurred o the die Probability that the ma does ot speak the truth Thus, by Bayes' theorem, we get P(S E) Probability that the report of the ma that six has occurred is actually a six Hece, the required probability is 3 8 Value based aswer Q.4.. E Items from Machie A ad E Items from Machie B Ehoosig a defective item P(E) 3/ P(E)/ P(E/E) 0 P(E/E) 00

7 P ( E / E ) P(E/ E P(E/ )P( E E )P( E ) P(E/ ) E )P( E ) mark Q.4. The ur cotais red ad black balls. Let a red ball be draw i the first attempt. P (drawig a red ball) If two red balls are added to the ur, the the ur cotais 7 red ad black balls. P (drawig a red ball) Let a black ball be draw i the first attempt. P (drawig a black ball i the first attempt) If two black balls are added to the ur, the the ur cotais red ad 7 black balls. P (drawig a red ball) Therefore, probability of drawig secod ball as red is Q. Let E ad E be the respective evets that a perso has a disease ad a perso has o disease. Sice E ad E are evets complimetary to each other, P (E) + P (E) P (E) P (E) Let A be the evet that the blood test result is positive.

8 Probability that a perso has a disease, give that his test result is positive, is give by P (E A). By usig Bayes theorem, we obtai Q.6. No. of questios Optio give i each questio 3 p probability of aswerig correct by guessig 3 q probability of aswerig wrog by guessig - p This problem ca be solved by biomial distributio. Where r four or more correct aswers 4 or P ( r ) r r c r 3 3 (i) P ( 4 ) c (ii) P ( ) c 3 P(4) + p() 4 c c

9 Q.7 : The repeated throws of a die are Beroulli trials. Let X deotes the umber of sixes i 6 throws of die. Obviously, X has the biomial distributio with 6 ad p /6 q -/6 /6 where p is probability of gettig a six ad q is probability of ot gettig a six Now, Probability of gettig at most sixes i 6 throws P (X 0) + P (X ) + P (X ) P q 6 P q 6 P q ! 6! 6!! 6 6! 4! Q.8 : p,q - 3 p 3 3 P ( r ) c q p r r r ; 6 P( r) 6 c r r r 3 3 P ( x 4 ) P ( x 4 ) P ( x ) P ( x 6 ) c 6 c 6 c Q.9: Let p deotes probability of gettig heads Let q deotes probability of gettig tails. p, q Suppose the coi is tossed times. Let X deotes the umber of times of gettig heads i trails.

10 r r r r P ( X r ) p q ( ) ( ) ( ), r 0,,,.... r r r P ( X ) P ( X ) P ( X ) P ( X ) P ( X 0 ) P ( X 0 ) 0 ( ) ( ) 3, 4,..... S o t h e f a i r c o i s h o u l d b e t o s s e d f o r 3 o r m o r e t i m e s f o r g e t t i g t h e r e q u i r e d p r o b a b i l i t y Q.0. As : j P i 0 + k + k + k + 3k + k + k + 7k + k 0k + 9k - 0 0k + 0k - k - 0 0k (k + ) - (k + ) 0 (k + ) (0k - ) 0 k - ad k /0 But k ca ever be egative as probability is ever egative. k /0 Now, k /0 P (X < 3) P (X 0) + P (X ) + P(X ) 0 + k + k 3k 3/0 3 P (X > 6) P (X 7) 7k + k.7/00 P (0 < X < 3) P (X ) + P (X ) k + k 3k 3/0 Q. : X ca take values 0,, P(X0) P(o card is red) 6/x / /0 P(X) P(oe card is red) 6/x6/+6/x6/ 6/ P(X) P(both cards are red) 6/x / /0 Mea 0(/0)+(6/)+(/0) 6/+x6/x/ Px i i

11 Variace Px - i i Q. Px i i The first six positive itegers are,, 3, 4,, 6. We ca select the two positive umbers i 6 30 differet ways. Out of this, umbers are selected at radom ad let X deote the larger of the two umbers. Sice X is the large of the two umbers, X ca assume the value of, 3, 4, or 6. P (X ) P (larger umber is ) {(,) ad (,)} /30 P (X 3) P (larger umber is 3) {(,3), (3,), (,3), (3,)} 4/30 P (X 4) P (larger umber is 4) {(,4), (4,), (,4), (4,), (3,4), (4,3)} 6/30 P (X ) P (larger umber is ) {(,), (,), (,), (,), (3,), (,3), (4,), (.4)} 8/30 P (X 6) P (larger umber is 6) {(,6), (6,), (,6), (6,), (3,6), (6,3), (4,6), (6,4), (,6), (6,)} 0/30 Give the above probability distributio, the expected value or the mea ca be calculated as follows: Mea (Xi P(Xi)) (Xi P(Xi)) /30+3 4/30+4 6/30+ 8/30+6 0/30 ( )/3040/304/3 HOTS Q. A ma is kow to speak the truth 3 out of times. He throws a die ad reports that it is a umber greater tha 4.fid the probability that it is actually a umber greater tha 4. (009) Q.oloured balls distributed i three bags as show i the followig table: Bag olour of the ball Black White Red I 3 II 4 III 4 3

12 A bag is selected at radom ad the two balls are radomly draw from the selected bag. They happe to be black ad red. What is the probability that they came from bag I? (009) Q.3 Give three idetical boxes I,II ad III each cotaiig two cois. I box I, both cois are gold cois, i box II both are silver cois ad i box III, there is oe gold coi ad oe silver coi. A perso chooses a box at radom ad takes out a coi. If the coi is of gold, what is the probability that the other coi i the box is also of gold? (0) Q.4 A bag cotais four balls. Two balls are draw at radom, ad are foud to be white. What is the probability that all ball are white? (00) Q. Suppose a girl throws a die. If she gets or 6, she tosses a coi 3 times ad otes the umber of heads. If she gets,,3 or 4 she tosses a coi oce ad ote whether a head or tail is obtaied. If she obtaied exactly oe head, what is the probability that she threw,,3, or 4 with the die? (0) Q.6 I a set of 0 cois, two cois are with heads o both the sides. A coi is selected at radom from this set ad tossed five times. If all the five times, the result was heads, fid the probability that the selected coi had heads o both the sides. (0) ANSWERS Q.: Let E gettig o. more tha 4 ; E gettig o. ot more tha 4 P(E) /3 ;P(E) /3 Let A be the evet perso is speakig truth P(A/E)3/ ; P(A/E)/ SO the probability for gettig a o. more tha 4 is P(E/A) P(E)P( A E P(E)P(A/E) )+P(E)P(A/E) 3/7 Q.: As bags are selected at radom P(bag I) /3 P(bag II) P(bag III) Let E be the evet that balls are black ad red. P x 3 x E E P b a g I 6 b a g I I 7 P E b a g I I I 4 x 3

13 We have to determie E P b a g I P b a g I b a g I P E E E E P b a g I P P b a g I I P P b a g I I I P b a g I b a g I I b a g I I I x 3 3 x x x Q.3: Let E, E, E3 be evets such that E Selectio of Box I ; E Selectio of Box II ; E3 Selectio of Box III Let A be evet such that A the coi draw is of gold Now, P (E) P (E ) P (E3 )/3 P ( A ) P(a gold coi from box I), P ( A ) P(a gold coi from box II) 0, E E P ( A ) (a gold coi from box III) / E3 the probability that the other coi i the box is also of gold P ( E A ) P(E)P(A/E) P(E)P ( A E ) + P(E)P ( A E ) + P(E3)P ( A E3 ) Q.4: Let us defie the followig evets, E: draw balls are white A: white balls i bag B: 3 white balls i bag : 4 white balls i bag The P(A) P(B) P() E E E P, P, P A 4 6 B By applyig Bayes theorem

14 P E P ( ) P E E E E P ( A ) P P ( B ) P P ( ) P A B Q. : Let E outcome or 6 ad E outcome of,,3,4 the P(E)/6/3 ; P(E)4/6/3 Let A be the evet of gettig oe head P(A E)3/8 P(A E) ½ ;the By usig Bayes Theorem P(E A) 8/ P(E).P(A E) P(E).P(A E)+P(E).P(A E) Q.6 Let E, E ad A be the evets defied as follows E selectig a coi havig head o both the sides E selectig a coi ot havig head o both the sides A gettig all heads whe a coi is tossed time P ( E ) 0 0 There are eight cois ot havig heads o both the sides P ( E ) P ( A / E ) ( ) P ( A / E ) ( ) 3 B y b a y e s t h e o r e m, w e h a v e P ( E / A ) P ( E ) P ( A / E ) P ( E ) P ( A / E ) P ( E ) P ( A / E )

15 HOT AND VALUE BASED QUESTION FOR SELF EVALUATION- Q.If each elemet of a secod order determiat is either 0 or.what is the probability that value of determiat is positve?also write dow the importace of positive thikig i yours daily life. As3/6 Q. P speaks truth 70 percet of the cases ad Q i 80 percet of the cases.i what percetage of cases they likely to agree i statig the same fact? Do you thik whe they agree, meas both are speakig truth? As: 3/0, o both ca tell a lie Q.3Fid the mea, the variace ad the stadard deviatio of the umber of doublets i three throws of a pair of dice? As: mea /, variace/ ad stadard deviatio Q.4.I a group of 0 scouts i a camp, 30 are well traied i first aid techiiques while the remaiig are well traied i hospitality but ot i first aid two scouts are selected at radom from the group fid the probability distributio of umber of selected scouts who are well traied i first aid.fid the mea of the distributio also. Write oe more value which is expected from a well traied scout As: Probability distributio X 0 P(X) 38/4 0/4 87/4 Mea94/4 Q. I a village there are 00 people out of them 70 people are o vegetaria. Two people are selected radomly. Fid the probability distributio of vegetaria people. Which type of people is better? Give yours opiio keepig i mid the importace of life of aimal i Eco system. As X 0 P(X) 483/990 40/990 87/990