Fift Fifty y Chal Challlengi engin ng Probability
Probl roble ems
in
1. A drawer drawer contains contains red socks and black socks. socks. When two socks socks are are drawn drawn at random, random, the probability probability that both are red is ½. (a) How small small can the the number number of socks socks in the drawer be? Ans be? Ans.. 4 (b) How small if the number of black socks is een? Ans. een? Ans. 21 !. "o encoura#e encoura#e $lmer%s promisin# tennis career, his father o&ers him a pri'e if he wins (at least) two tennis sets in a row in a threeset series to be played with his his fath father er and and the the club club cham champi pion on alternately alternately fatherchampionfa fatherchampionfather ther or championfather championfatherchampion, champion, accordin# accordin# to $lmer%s choice. choice. "he champion champion is a better player than $lmer%s father. Which series should $lmer choose? Ans. Champion-Father-Champion (0.512) Father-Champion-Father (0.384) *. A three threem man an +ury +ury has has two membe members rs each each of whom whom inde indepe pend nden entl tly y has has prob probab abil ilit ity y p of maki makin# n# the the corr correc ectt decision and a third member who ips a coin for each decision (ma+ority rules). A oneman +ury has probability p probability p of of makin# the correct decision. Which +ury has the better probability of makin# the correct decision? Ans. Both Both one-ma one-man n jury jury and threethreeman jury -. n the aera#e, how many times must a die be thrown until one #ets a /?
Ans. 6 Ans. !ind Friend "i## be $ured %r&t 0. n a common carnial carnial #ame, a player player tosses a penny from a distance of about 0 feet onto the surface of a table ruled in 1inch s2uares. f the penny (*3- inch in diameter) falls entirely inside a s2uare, the player receies 0 cents but does not #et his penny back4 otherwise he loses his penny penny. f the the penn penny y lands lands on the the table, what is his chance to win?
. We often often read of someon someone e who has been dealt 1* spades at brid#e. With a wellshu@ed pack of cards, what is the chance that you are dealt a perfect hand (1* of one suit)? (:rid#e is played with an ordinary pack of 0! cards, 1* in each of - suits, and each of - players is dealt 1*.)
Ans. 1/16 Ans. 52 ' 12 /. 5hucka6uck is a #amblin# #ame often play played ed at carn carni ial als s and and #amb #ambli lin# n# houses. A player may bet on any one of the numbers , !, *, -, 0, /. "hree dice are are rolle olled. d. f the the play player er%s %s numb number er appear appears s on one, two, or thre three e of the the dice, he receies respectiely one, two, or three times his ori#inal stake plus his own money back4 otherwise he loses his stake. stake. What is the player%s player%s e7pecte e7pected d loss per unit stake? (Actually the player may may dist distri ribu bute te stak stakes es on see seera rall numbers, but each such stake can be re#arded as a separate bet.) Ans. 8 8. 9r. :rown :rown always bets a dollar dollar on the number 1* at roulette a#ainst the adice of ;ind at een een money money that :rown :rown will be behind at the end of */ plays. How How is the cure ure worki orkin n#? (9ost ost Amer Americ ican an roul roulet ette te whee wheels ls hae hae * e2ually e2ually likely likely numbers. numbers. f the player%s player%s number comes up, he is paid *0 times his stake stake and #ets his ori#in ori#inal al stake stake back4 otherwise he loses his stake.)
. "he #ame of craps craps,, played played with two dice, dice, is one one of Ameri America% ca%s s fastes fastestt and most ost popu opular lar #amb amblin lin# #ame ames. 5alculatin# the odds associated with it is an instructie e7ercise. "he rules are thes these. e. nly nly tota totals ls for for the the two two dice dice count. "he player throws the dice and wins at once if the total for the Brst throw is 8 or 11, loses at once if it is !, *, or 1!. Any other throw is called his CpointD. f the Brst throw is a point, the player throws the dice repeatedly until he eithe eitherr wins wins by thro throwin win# # his point point a#ain or loses by throwin# 8. What is the player%s chance to win? Ans. 0.42* 1>. (a) An urn contains 1> black balls and 1> white balls, identical e7cept for colo colorr. Eou choo choose se Cbla Cblack ckCC or Cwhi Cwhite te.C .C ne ne ball ball is draw drawn n at random, and if its color matches yo ur ur choice, you #et =1>, otherwise nothin#. nothin#. Write down the ma7imum amount you are willin# to pay pay to play play the the #ame. ame. "he "he #ame will be played +ust once.
Ans. +ou &hou#d be "i##in, to pay at #ea&t a& mu$h to p#ay the &e$ond ,ame a& the %r&t. Furthermore i you hae any inormation about your riend& preeren$e& you $an tae adanta,e o that to improe your $han$e&.
not chosen a meetin# place, and neither can reach the other because both hae embarked on trips. f they try neertheless to meet, where should they #o? Ans. mpire 3tate Bui#din,
10. $i#ht eli#ible bachelors and seen beautiful models happen randomly to hae purchased sin#le seats in the same 1 0seat row of a theater. n the aera#e, how many pairs of ad+acent seats are ticketed for marria#eable couples? Ans. /15
(b) A friend of yours has aailable many black and many white balls, and he puts black and white balls into the urn to suit himself. Eou choose CblackC or Cwhite.C A ball is drawn randomly from this urn. Write down the ma7imum amount you are willin# to pay to play this #ame. "he #ame will be played +ust once. Ans. +ou &hou#d be "i##in, to pay at #ea&t a& mu$h to p#ay the &e$ond ,ame a& the %r&t. Furthermore i you hae any inormation about your riend& preeren$e& you $an tae adanta,e o that to improe your $han$e&. 11. "wo stran#ers are separately asked to choose one of the positie whole numbers and adised that if they both choose the same number, they both #et a pri'e. f you were one of these people, what number would you choose? Ans. 1 1!. "wo stran#ers who hae a priate reco#nition si#nal a#ree to meet on a certain "hursday at 1! noon in Few Eork 5ity, a town familiar to neither, to discuss an important business deal, but later they discoer that they hae
1*. "hree prisoners, A, :, and 5, with apparently e2ually #ood records hae applied for parole. "he parole board has decided to release two of the three, and the prisoners know this but not which two. A warder friend of priso ner A knows who are to be released. Grisoner A reali'es that it would be unethical to ask the warder if he, A, is to be released, but thinks of askin# for the name of one prisoner other than himself who is to be released. He thinks that before he asks, his chances of release are !3*. He thinks that if the warder says C: will be released,C his own chances hae now #one down to 13!, because either A and : or : and 5 are to be released. And so A decides not to reduce his chances by askin#. Howeer, A is mistaken in his calculations. Ans. robabi#ity o & re#ea&e 7 2/* 1-. 5oupons in cereal bo7es are numbered 1 to 0, and a set of one of each is re2uired for a pri'e. With one coupon per bo7, how many bo7es on the aera#e are re2uired to make a complete set? Ans. 11.42
1/. A tennis tournament has players. "he number a player draws f rom a hat decides his Brstround run# in the tournament ladder. uppose that the best player always defeats the ne7t best and that the latter always defeats all the rest. "he loser of the Bnals #ets the runnerup cup. What is the chance that the secondbest player wins the runnerup cup? Ans. 4/ 18. (a) uppose ;in# Arthur holds a +oustin# tournament where the +ousts are in pairs as in a tennis tournament. ee Groblem 1/ for tournament ladder. "he kni#hts in the tournament are eenly matched, and they include the twin kni#hts :alin and :alan. What is the chance that the twins meet in a match durin# the tournament? Ans. 1/4 (b) Ieplace by !n in the aboe problem. Fow what is the chance that they meet?
Ans. 1/2n-1 1. When 1>> coins are tossed, what is the probability that e7actly 0> are heads? Ans. 0.08
1>> blacks. An urn is chosen at random, and you win a pri'e if you correctly name the urn on the basis of the eidence of two balls drawn from it. After the Brst ball is drawn and its color reported, you can decide whether or not the ball shall be replaced before the second drawin#. How do you order the second drawin#, and how do you decide on the urn? Ans. 9hereore i red i& dra"n %r&t rep#a$e it beore dra"in, a,ain
1. Gepys wrote Fewton to ask which of three eents is more likely that a person #et (a) at least si7 when / dice are rolled, (b) at least ! si7es when 1! dice are rolled, or (c) at least * si7es when 1 dice are rolled. What is the answer? Ans. epy& "i## do better "ith the 6di$e "a,er than "ith 12 or 18 !>. A, :, and 5 are to B#ht a three cornered pistol duel. All know that A%s chance of hittin# his tar#et is >.*, 5%s is >.0, and : neer misses. "hey are to Bre at their choice of tar#et in succession in the order A, :, 5, cyclically (but a hit man loses further turns and is no lon#er shot at) until only one man is left unhit. What should A%s strate#y be? Ans. 3o %re& hi& %r&t &hot into the ,round and then trie& to hit B "ith hi& ne't &hot !1. "wo urns contain red and black balls, all alike e7cept for color. Jrn A has ! reds and 1 black, and Jrn : has 1>1 reds and
!!. n an election, two candidates, Albert and :en+amin, hae in a ballot bo7 a and b otes respectiely, a K b, for e7ample, * and !. f ballots are randomly drawn and tallied, what is the chance that at least once after the Brst tally the candidates hae the same number of tallies? Ans. 0.08 !*. Glayers A and : match pennies F times. "hey keep a tally of their #ains and losses. After the Brst toss, what is the chance that at no time durin# the #ame will they be een? N
Ans.
n N
2
!-. 9arin #ets o& work at random times between * and 0 G.9. His mother lies uptown, his #irlfriend downtown. He takes the Brst subway that comes in either direction and eats dinner with
the one he is Brst deliered to. His mother complains that he neer comes to see her, but he says she has a 0>0> chance. He has had dinner with her twice in the last !> workin# days. Ans .
1 minute intera# bet"een a
do"nto"n and an upto"n train !0. f a chord is selected at random on a B7ed circle, what is the probability that its len#th e7ceeds the radius of the circle? (a) Assume that the distance of the chord from the center of the circle is eenly (uniformly) distributed from > to r.
Ans.
√ 3 / 2 : 0.866
(b) Assume that the midpoint of the chord is eenly distributed oer the interior of the circle. Ans. */4 7 0.5 (c) Assume that the chord is determined by two points chosen so that their positions are independently eenly distributed oer the circumference of the ori#inal circle. Ans. 2/* !/. Luels in the town of Liscretion are rarely fatal. "here, each contestant comes at a random moment between 0 a.m. and / a.m. on the appointed day and leaes e7actly 0 minutes later, honor sered, unless his opponent
arries within the time interal and then they B#ht. What fraction of duels lead to iolence? Ans. 1/6
Ans. 0.224
!8. (a) "he kin#%s minter bo7es his coins 1>> to a bo7. n each bo7 he puts 1 false coin. "he kin# suspects the minter and from each of 1>> bo7es draws a random coin and has it tested. What is the chance the minter%s peculations #o undetected? Ans. 0.*66 (b) What if both 1%s are replaced by n?
Ans.
1
!. "he kin#%s minter bo7es his coins n to a bo7. $ach bo7 contains m false coins. "he kin# suspects the minter and randomly draws 1 coin from each of n bo7es and has these tested. What is the chance that the sample of n coins contains e7actly r false ones? −m
Ans.
m
*>. A bread salesman sells on the aera#e !> cakes on a round of his route. What is the chance that he sells an een number of cakes? (We assume the sales follow the Goisson distribution.)
r
r!
!. Airborne spores produce tiny mold colonies on #elatin plates in a laboratory. "he many plates aera#e * colonies per plate. What fraction of plates has e7actly * colonies? f the
*-. 6abor laws in $rewhon re2uire factory owners to #ie eery worker a holiday wheneer one of them has a birthday and to hire without discrimination on #rounds of birthdays. $7cept for these holidays they work a */0day year. "he owners want to ma7imi'e the e7pected total number of mandays worked per year in a factory. How many workers do factories hae in $rewhon? Ans. *64
Ans. 0.568 *1. What is the least number of persons re2uired if the probability e7ceeds 13! that two or more of them hae the same birthday? (Eear of birth need not match.)
n
( n) −
1
e
aera#e is a lar#e inte#er m, what fraction of plates has e7actly m colonies?
Ans. 2* *!. Eou want to Bnd someone whose birthday matches yours. What is the least number of stran#ers whose birthdays you need to ask about to hae a 0>0> chance? Ans. 25*
**. f r persons compare birthdays in the pairin# problem, the probability is PR that at least ! hae the same birthday. What should n be in the personal birthmate problem to make your probability of success appro7imately PR?
Ans.
r ( r −1 ) 2
*0.
*8. At 6as Me#as, a man with =!> needs =->, but he is too embarrassed to wire his wife for more money. He decides to inest in roulette (which he doesn%t en+oy playin#) and is considerin# two strate#ies bet the =!> on CeensC all at once and 2uit if he wins or loses, or bet on CeensC one dollar at a time
until he has won or lost =!>. 5ompare the merits of the strate#ies. Ans. f he beats =!> at once and 2uits #ies him the probability of 0.44 f he beats =1 at a time until he has won =!> #ies him the probability of 0.11 *. How thick should a coin be to hae a 13* chance of landin# on ed#e? Ans. *5 a& thi$ a& the diameter o $oin *. n a laboratory, each of a handful of thin inch #lass rods had one tip marked with a blue dot and the other with a red. When the laboratory assistant tripped and dropped them onto the concrete oor, many broke into three pieces. . hu@e an ordinary deck of 0! playin# cards containin# four aces. "hen turn up cards from the top until the Brst ace appears. n the aera#e, how many cards are re2uired to produce the Brst ace?
Ans. 11 (b) Eou hae looked at 0 locomoties and the lar#est number obsered is />. A#ain #uess how many locomoties the company has.
win the #ame you must #et more than half the points. Eou know p, say >.-0, and you #et a pri'e if you win. Eou #et to choose in adance the number of plays. How many do you choose? Ans. 10
Ans. 8 -!. (a) f a stick is broken in two at random, what is the aera#e len#th of the smaller piece? Ans. 1/4 of its len#th (b)What is the aera#e ratio of the smaller len#th to the lar#er? Ans. Iatio is 0.*86
-0. (a)
-*. A bar is broken at random in two places.
(b) A typist types letters and enelopes to n di&erent persons. "he letters are randomly put into the enelopes. n the aera#e, how many letters are put into their own enelopes? Ans. 1 -/. Jnder the conditions of the preious matchin# problem, what is the probability of e7actly r matches?
P ( r│ n ) ≈ 1 e− r!
1
Ans. 10.6th $ard
-1. (a) A railroad numbers its locomoties in order, 1, !, ... , F. ne day you see a locomotie and its number is />. Nuess how many locomoties the company has.
Ans.
--. A #ame consists of a se2uence of plays4 on each play either you or your opponent scores a point, you with probability p (less than 13!) he with probability 1 p. "he number of plays is to be een ! or - or / and so on. "o
-8. "he kin#, to test a candidate for the position of wise man, o&ers him a chance to marry the youn# lady in the court with the lar#est dowry. "he
amounts of the dowries are written on slips of paper and mi7ed. A slip is drawn at random and the wise man must decide whether that is the lar#est dowry or not. f he decides it is, he #ets the lady and her dowry if he is correct4 otherwise he #ets nothin#. f he decides a#ainst the amount written on the Brst slip, he must choose or refuse the ne7t slip, and so on until he chooses one or else the slips are e7hausted. n all, 1>> attractie youn# ladies participate, each with a di&erent dowry. How should the wise man make his decision? Ans.
1
/e
-. As a second task, the kin# wants the wise man to choose the lar#est number from amon# 1>>, by the same rules, but this time the numbers on the slips are randomly drawn from the numbers from > to 1 (random numbers, uniformly distributed). Fow what should the wise man%s strate#y be? Ans.
/
3 4
-. An unbiased instrument for measurin# distances makes random errors whose distribution has standard deiation u. Eou are allowed two measurements all told to estimate the len#ths of two cylindrical rods, one clearly lon#er than the other. 5an you do better than to take one measurement on each rod? (An unbiased instrument is one that on the aera#e #ies the true measure.)
Ans. +e& 0>.
What is the probability that the 2uadratic e2uation x 2 + 2 bx + c = 0 has real roots?
Ans. 1 01. tartin# from an ori#in >, a particle has a 0>0> chance of moin# 1 step north or 1 step south, and also a 0>0> chance of moin# 1 step east or 1 step west. After the step is taken, the moe is repeated from the new position and so on indeBnitely. What is the chance that the particle returns to the ori#in? Gart of lattice of points traeled by particles in the twodimensional random walk problem. At each moe the particle #oes one step north east, northwest, southeast, or southwest from its current position, the directions bein# e2ually likely. What is the chance that the particle returns to the ori#in?
0*. A table of inBnite e7panse has inscribed on it a set of parallel lines spaced 2a units apart. A needle of len#th 2 (smaller than 2a) is twirled and tossed on the table. What is the probability that when it comes to rest it crosses a line? Ans. 1 0-. uppose we toss a needle of len#th 2 (less than 1) on a #rid with both hori'ontal and ertical rulin#s spaced one unit apart. What is the mean number of lines the needle crosses? ( hae dropped the !a for the spacin# because we mi#ht as well think of the len#th of the needle as measured in units of spacin#.) Ans. 1.2
Ans. 1 0!. As in the twodimensional walk, a particle starts at an ori#in > in three space. "hink of the ori#in as centered in a cube ! units on a side. ne moe in this walk sends the particle with e2ual likelihood to one of the ei#ht comers of the cube. "hus, at eery moe the particle has a 0>0> chance of moin# one unit up or down, one unit east or west, and one unit north or south. f the walk continues foreer, Bnd the fraction of particles that return to the ori#in. Ans. 0.2*
00. n the preious problem let the needle be of arbitrary len#th, then what is the mean number of crosses? 4
Ans.
( originallength )
π
drawin#s and the composition of the two urns so that the probability that all white balls are drawn from the Brst urn is e2ual to the probability that the drawin# from the second is either all whites or all blacks. 0/. "wo urns contain the same total numbers of balls, some blacks and some whites in each.
Ans. zn = x n + y n Where ! be the number of white balls in the Brst urn4
x the number of whites and4 " the number of blacks in the second Atho#$h% &or man" 'a#es o& n% it is non that this e*#ation cannot be satise,% it is not non hether it is impossibe &or a 'a#es o& n3. #t it is non to be impossibe &or n < 2000.
