Question 1:
Sri Lanka needs 12 runs in the last four balls and Jayasuriya is batting. Assume that he does not get out, and either hits a six or misses each ball. (1a) hat is the !robability that they in o" the last ball# (1b) hat is the !robability that they in#
Question 2:
$r. %arib &han, $arketing $anager in'charge of Soft hite detergent is ex!loring the !ossib !ossibili ility ty of ader adertis tising ing his his !rodu !roduct ct on the u!comi u!coming ng ham!i ham!ions ons** +ro!hy ro!hy +ournament +ournament of -s. /is adertising agency agency has informed him that the regular regular slots (beteen oers) hae already been taken, but 0s!ecial slots are still aailable. +hese are the slots aailable hen ickets fall. ith his long ex!erience, he knos that the ieers* ieers* intensit intensity y is strong strong during during the 3rst hour that ndia is batting. batting. f ndia is batting and 4 or more of ickets fall in the 3rst hour, Soft hite ill hae /igh 5x!osure 5x!osure (0/o un!atriotic, un!atriotic, but that is life, he told himself). himself). f less than 3e ickets fall fall during the 3rst hour, hour, Soft hite ill hae Lo Lo ex!osure. ex!osure. /e also knos that the !robabilities di"er de!ending on hether ndia is batting 3rst or 3elding 3rst. /e has estimated that the !robability !robability of 4 or more more ndian ickets ickets falling, hen ndia ndia is battin batting g 3rst 3rst is 6.76. 6.76. n the other other hand if ndia ndia is batti batting ng second second (i.e., (i.e., 3elding 3rst) then the the !robability !robability of 4 or more ickets ickets falling is higher higher at 6.86. /e rightly assumed that the !robability of ndia inning the toss is 6.4. f course, hether ndia is bating 3rst or 3elding 3rst de!ends on ndia inning the toss, hich is com!letely unbiased. unbiased. 9ast 9ast data indicates that in similar tournaments, tournaments, ndia ndia elected to bat :6; of the time after after inning inning the toss. n the other hand, hand, ndia as !ut to bat only <6; of the time hen they lost the toss. (2a) hat is the =oint !robability that ndia ins the toss AND elects to bat 3rst# (2b) hat is the =oint !robability that ndia loses the toss AND elects to bat 3rst# (2c) hat is the !robability that ndia bats 3rst# Question 3:
n a !recision bombing attack there is a 46; chance that any one bomb ill strike the target. +o direct hits are re>uired to destroy the target com!letely. /o many bombs must be dro!!ed to gie a ??; chance or better of com!letely destroying the target# Question 4:
Sarin !layed a large number of one'day cricket matches for the home team and he alays came in to bat at number three. As in the case of many other !layers, it as felt that his !erformance !erformance as not ery consistent. consistent. /is coach analy@ed his batting batting and found found a ery interesti interesting ng !attern. !attern. /e found that that Sarin came in to bat in the
early oers (3rst 1 oers) 46; of the timeB in the middle oers (beteen 18 th and <
Soura is assigned to lead a !ro=ect hich is scheduled to be com!leted in 7 eeks. /e knos that he ould =ust be able to com!lete the !ro=ect on time !roided all < team members (excluding him) stay on till the com!letion of the !ro=ect. hile Soura knos that there is no >uestion of him >uitting during the !ro=ect, he cannot be certain about any of his team members. ndeed he estimates, going by the industry !ractice, anybody has 16; chance of >uitting on any gien eek. hile he can and ill seek immediate re!lacement if any of the team'members leae during the !ro=ect, (and getting re!lacement is not a !roblem) naturally there is some delay in the !ro=ect com!letion de!ending on the !rogress status of the !ro=ect. om!any !olicy s!eci3es that a !erson can leae only at the end of a eek and for sim!licity you may assume that this !ro=ect started on $onday. +hus, for exam!le, there is no >uestion of any of the < initial co'members leaing during eek 1 although any number of them may leae at the end of eek 1 and hence not aailable from eek 2 of the !ro=ect. are is taken hile allotting re!lacements to ensure, for exam!le, that if there is any re!lacement at the beginning of eek 2, these re!lacements can leae earliest at the end of eek 2 etc. Soura gauges that there ould be a 1'day delay for eery team'member leaing at the end of the 1 st eek of the !ro=ect, and 2'day delay for eery member leaing at the end of the second eek , and <'day delay !er member leaing at the end of eek<. (4a)hat is the !robability distribution of the number of team members of Soura leaing at the end of eek 1# (4b)hat is the !robability of the !ro=ect getting com!leted on time# (4c)hat is the chance that the delay in com!letion (if any) ill be at most 2 days# (4d)%ien that at least 1 team'member leaes at the end of eek 2 as ell as at the end of eek <, hat is the likelihood that the delay in !ro=ect com!letion ill be at most days# Question 6:
n a book of 4?6 !ages, 6 ty!o'gra!hical errors occur. Assuming 9oisson la for the number of errors !er !age, 3nd the !robability that a random sam!le of 4 !ages ill contain no error. Question 7:
Su!!ose that the number of tele!hone calls coming into a tele!hone exchange beteen 16am and 11 am is a random ariable hich is 9oisson distributed ith !arameter 2. Similarly the number of calls arriing beteen 11am to 12 noon is another 9oisson distributed random ariable ith !arameter . f the to random ariables are inde!endent, hat is the !robability that more than 4 calls come in beteen 16 am to 12 noon# Question 8:
Subbu has recently =oined Setha Enlimited Serices ndia (Ltd) also knon as SESL. SESL is a ma=or serice !roider to many softare com!anies in Cangalore. SESL handles the salary !ayments as ell as the salary deductions to the em!loyees of the softare com!anies. +hey also hae to handle the com!laints receied from the em!loyees. Subbu is one of the 0com!laint handlers s!ecially designated to handle the com!laints of the em!loyees of ndosys 9t Ltd. +he number of com!laints that he handles in a day is random but on aerage, he handles : com!laints !er day. SESL, based on their contract ith ndosys, !ays Subbu according to his !erformance. /is !erformance is rated as 0Lo, if he handles 4 or less number of com!laints in a day. /is !erformance is rated as 09ar if he handles to 16 com!laints in a day and 05xcellent if he handles more than 16 com!laints in a day. /e is !aid Fs. 266 !er a 0Lo day, Fs. <66 !er a 09ar day and Fs. 66 !er an 05xcellent day. Subbu feels this is somehat unfair, since the arrial of com!laints is a random !rocess (9oisson) oer hich he has no control. /oeer, he has little choice in the matter. (:a)n a !articular day selected at random, hat is the !robability that Subbu is !aid Fs. 66# (:b)n a 4'day eek, hat is the !robability that Subbu has < 0Lo days# (:c)hat is the aerage amount receied by Subbu in a day# Question 9:
An automatic 3lling machine 3lls cola into 846 ml bottles. At the start of a shift an ins!ector ill take a random sam!le of 46 bottles and measure the olume of cola in them accurately. After com!uting the sam!le mean, she ill a!!roe the machine o!eration only if the sam!le mean falls beteen 87?.4 and 846.4 ml. f she does not a!!roe, the maintenance cre ill be called to check and rectify the machine setu!. Assume that during a !articular shift the !o!ulation mean as 846.7 ml and the !o!ulation standard deiation as 1.2 ml. hat is the !robability that the ins!ector ill a!!roe the machine o!eration#
Question 10: X 1 , X 2 ,.........
Let
be i.i.d. (inde!endent and identically distributed) 9oisson random λ = 2 ariables ith !arameter . Ese entral Limit +heorem to estimate P (120 ≤ S n ≤ 160 ) S n = X 1 + X 2 + ............... + X n n = 75 . e also hae B