PROBABILITY
30.6 X 1 ,X 2 ,...,Xnare independent, identically distributed, random variables drawn
from a uniform distribution on [0,1]. The random variablesAandBare defined
by
A=min(X 1 ,X 2 ,...,Xn),B= max(X 1 ,X 2 ,...,Xn).
For any fixedksuch that 0≤k≤^12 , find the probability,pn, that both
A≤k and B≥ 1 −k.Check your general formula by considering directly the cases (a)k=0,(b)k=^12 ,
(c)n=1and(d)n=2.
30.7 A tennis tournament is arranged on a straight knockout basis for 2nplayers,
and for each round, except the final, opponents for those still in the competition
are drawn at random. The quality of the field is so even that in any match it is
equally likely that either player will win. Two of the players have surnames that
begin with ‘Q’. Find the probabilities that they play each other
(a) in the final,
(b) at some stage in the tournament.30.8 This exercise shows that the odds are hardly ever ‘evens’ when it comes to dice
rolling.
(a) GamblersAandBeach roll a fair six-faced die, andBwins if his score is
strictly greater thanA’s. Show that the odds are 7 to 5 inA’s favour.
(b) Calculate the probabilities of scoring a totalTfrom two rolls of a fair die
forT=2, 3 ,...,12. GamblersCandDeach roll a fair die twice and score
respective totalsTCandTD,Dwinning ifTD>TC. Realising that the odds
are not equal,Dinsists thatCshould increase her stake for each game.C
agrees to stake£1.10 per game, as compared toD’s£1.00 stake. Who will
show a profit?30.9 An electronics assembly firm buys its microchips from three different suppliers;
half of them are bought from firmX, whilst firmsYandZsupply 30% and
20%, respectively. The suppliers use different quality-control procedures and the
percentages of defective chips are 2%, 4% and 4% forX,YandZ, respectively.
The probabilities that a defective chip will fail two or more assembly-line tests are
40%, 60% and 80%, respectively, whilst all defective chips have a 10% chance
of escaping detection. An assembler finds a chip that fails only one test. What is
the probability that it came from supplierX?
30.10 As every student of probability theory will know, Bayesylvania is awash with
natives, not all of whom can be trusted to tell the truth, and lost, and apparently
somewhat deaf, travellers who ask the same question several times in an attempt
to get directions to the nearest village.
One such traveller finds himself at a T-junction in an area populated by the
Asciis and Bisciis in the ratio 11 to 5. As is well known, the Biscii always lie, but
the Ascii tell the truth three quarters of the time, giving independent answers to
all questions, even to immediately repeated ones.
(a) The traveller asks one particular native twice whether he should go to the
left or to the right to reach the local village. Each time he is told ‘left’. Should
he take this advice, and, if he does, what are his chances of reaching the
village?
(b) The traveller then asks the same native the same question a third time, and
for a third time receives the answer ‘left’. What should the traveller do now?
Have his chances of finding the village been altered by asking the third
question?