Technical questions 27 * 29. 27*30 – 27 = 810 – 27 = 783. 19 * 19. 361, 20 * 20 – 20 – 19 = 400 – 39 = 361. 17 * 3, then *3, then *3, then *3, then *3. 17 * 3 = 51. 51 * 3 = 153. 153 * 3 = 459. 459 * 3 = 450 * 3 + 27 = 1377. 1377 * 3 = 3900 + 3 * 77 = 3900 + 210 + 21 = 4131. 79^2. 80 * 80 – 80 – 79 = 6400 – 159 = 6241. 39^2. 40 * 40 – 40 – 39 = 1600 – 79 = 1521. 29^2. 30 * 30 – 30 – 29 = 841. 19 * 21. (20 – 1)(20 + 1) = 400 – 1 = 399. 17 * 13. (15 + 2)(15 – 2) = 225 – 4 = 221. 27 * 33. (30 – 3)(30 + 3) = 900 – 9 = 891. What is 92 divided by 9? 10 + 2/9 What is the sum of 1!"? 41 * 20 = 820. What is the squa#e #oot of 2"""? sqrt(2000) = 10 * sqrt(20) which is slightly lss th!" 45. $% of a dice? 3.5. What&s bi''e#, e (i o# (ie? #i, $%r "&'rs gr!tr th!" 2 , iggr #%""t '!ttrs '%r. )f you have a sided cube that has a unique colo# on each side + diffe#ent colo#s, homany (ossible combinations of unique cubes a#e the#e? 3 #!irs %$ %##%sit c%l%rs, $irst #!ir is 6 ch%%s 2 = 15 w!ys, sc%" #!ir is 4 ch%%s 2 = 6 w!ys, thir #!ir is 1 w!y. -rr -rr %$ #!irs %"t '!ttr '!ttr s% ii ii y 3 = 15 * 6 / 6 = 15 w!ys. w!ys. !ch #!ir #!ir c!" r!rr!"g s% 15 * 2 = 30 w!ys. What is the chance of d#a-in' t-o ca#ds f#om the to( of the dec that tu#ns out to be the same numbe#? 1 * 3/51. - - :;-:< ->;?@A !"gl = B 0.5(60 C – 11?) B What is the an'le bet-een the hou# and minute hand at 3/3"? 30 grs #r h%&r $%r th h%&r h!", s% 30 + 30 + 15 = 75 grs. What is the an'le bet-een the hou# and minute hand at 3/10? 30 grs #r h%&r $%r th h%&r h!", s% 15/60 * 30 = 7.5 grs. What is the an'le bet-een the hou# and minute hand at 0/12? 12/60 is 1/5, s% h%&r h!" is 6 grs #!st 5#'. !ch 'i"&t is 6 grs. @% 5 * 30 + 6 – 72 = 84 grs. What is the sum of the di'its of you# (hone numbe#? qua#e it. Then divide it by 0. 304D553D 4973. 7 + 10 + 3 = 20, 20 + 10 + 13 = 43. 43 * 43 = 40 * 43 + 129 = 1849. 1849 / 5 = 369.8. What is the e(ected value of a dec of ca#ds? g"%r s&its. r!g %$ 1 !" 13 is 7. o- many 0ca#d (oe# hands a#e the#e that mae ! of a ind? 13 i$$r"t 4 %$ ! Ei"s, 5th c!r c!" %" %$ 48 %thr c!rs. 13 * 48 = 650 – 26 = 624. What is a 4a#ov 5hain? st%ch!stic syst' i" which th c&rr"t st!t %"ly #"s %" th %" #ri%&s st!t. ls% E"%w" !s !" !&t%rgrssiD1 '%l. What is the chance that at least t-o (eo(le -e#e bo#n on the same day of the -ee if the#e a#e 3 (eo(le in the #oom? 1 – ch!"c th!t "%%y is %r" %" th s!' !y %$ th wE = 1 – (1)(6/7)(5/7) = 19/49 T-o #o(es a#e nonunifo#m in com(osition. They bu#n com(letely in 1 hou#. 6ou only have a li'hte# and those t-o #o(es and you have to tell -hen !0 minutes have (assed. ;ight %th "s %$ %" r%#, !" %" " %$ th %thr r%#. $tr th $irst r%# &r"s %&t, 30 'i"&ts will h! #!ss, !" light th %thr " %$ th sc%" r%#. Fh" th sc%" r%# &r"s %&t, 15 '%r 'i"&ts will h! #!ss, t%t!li"g 45 'i"&ts. ) have a a# filled -ith 1"" (ennies. 1 of the (ennies has t-o heads, and the othe#s a#e no#mal (ennies. ) #andomly choose one (enny, fli( it 7 times, and 'et 7 heads. 8m ) holdin' • • •
• • • • • • • • • • • •
•
• •
•
•
•
• •
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
the t-oheaded (enny o# a no#mal (enny? :h!"c %$ th tw%Dh! #""yA 1/100. :h!"c %$ gtti"g ! "%r'!l #""y !" $li##i"g s" h!s 99/100 * 1/128. G%& h! ! highr ch!"c %$ h!i"g th tw%Dh! #""y. ) am thinin' of a numbe# that does not contain the di'it 1. The (#oduct of the di'its is 9. What is the smallest (ossible numbe# in my mind? o- about the la#'est? 96 is 2H5 * 3. ;!rgst is 322222. G%& c!" gr% thr 2s t%gthr !" !"%thr 2 !" 3 t%gthr i"t% 8 !" 6, rs#ctily. hr is %" l!st 2 l$t, s% y%& '!E th s'!llst "&'r with 8, 6, !" 2, which is 268. laye# 1 chooses a numbe#, then (laye# 2 chooses a diffe#ent one. 2 dice a#e tossed, and -hoeve# is close# to the sum -ins. Would you be 1 o# 2? What numbe# do you choose? > #l!yr 1, si"c th st ch%ic is th "&'r 7. 6ou #oll a die, and ) 'ive you that numbe# in dolla#s. 6ou can acce(t at any time but have a ma of 3 #olls and must acce(t the thi#d. o- do you maimi:e you# #etu#n? #ct !l& %$ 3r r%ll is 3.5. @% #icE sc%" i i$ 4, 5, %r 6. #ct !l& %$ 2 " !" 3 r r%lls is ((4 + 5 + 6)/3 + 3.5)/2 = 4.25, s% #icE 5 %r 6 $%r 1 st r%ll. What is the e(ected value to you of the follo-in' 'ame? 6ou and anothe# (e#son have 3 coins each. 6ou both fli( all of you# coins and if you# th#ee coins sho- the same numbe# of tails as the othe# (e#son, you (ay the ;2, othe#-ise they (ay you ;1. r%!ility th!t y%& l%s is (1/8)H2 + (1/8)H2 + (3/8)H2 + (3/8)H2 = 20/64 = 5/16. (D2)*(5/16) + (11/16) = 1/16. 6ou have 2<3 chance of hittin' a ta#'et. What is the chance of hittin' it if you fi#e t-ice? 1 – "% hits = 1 – (1/3)(1/3) = 8/9. 6ou see that the ta#'et has been hit. What is the chance that you hit it t-ice? r%!ility %th hit / #r%!ility y%& hit !t !ll = (4/9) / (8/9) = 1/2. The#e a#e 99 lions and 1 shee( on an island. =ions -ant to eat shee( but also -ant to stay alive. When a lion eats a shee(, it tu#ns into a shee(. o- many lions and shee( left a#e left afte# a (e#iod of time? +=ions can su#vive on othe# foods on the island, shee( cannot esca(e lions, and all c#eatu#es a#e #ational $ thr wr 2 li%"s, !" 1 sh#, th li%"s w%&l"t !t c!&s th %thr li%" w%&l !t th tr!"s$%r' li%". C%wr i$ thr wr 3 li%"s, th 3 r li%" w%&l !t th sh# E"%wi"g th #ri%&s $!ct. :%"ti"& !" "%tic th!t % li%"s !t !" " %"s %"t. @% th l%"gDtr' #ct "&'rs %$ li%"s !" sh# !r 98 !" 1, rs#ctily. 6ou #oll t-o dice. What is the (#obability of #ollin' a >3 total befo#e #ollin' a >7 total? r%!ility %$ 3 is 1/18. r%!ility %$ 7 is 1/6. 1 – 1/18 – 1/6 = 7/9. = (1/18) + (7/9), = 1/4. What is the (#obability that -hen you a##an'e 0 (eo(le in a #ound table, they a#e in inc#easin' a'e o#de# +cloc-ise o# anticloc-ise? 2 #%ssil !rr!"g'"ts (i"cr!si"g %r cr!si"g), 5 st!rti"g #%siti%"s, 5 c%'i"!ti%"s. 2*5/5 = 10/120 = 1/12. )f a 000 @ubi&s cube is sittin' on a des, eactly ho- many cubes have only one side touchin' the ai#? 9 * 5 = 45. )f you have a dec of ca#ds s(lit into ! (iles and -as offe#ed 1/1 odds to d#a- a face ca#d +A B C 8 f#om at least one of the (iles, -ould you tae the 'ame? Why o# -hy not? I%"t r!w ! c!r $r%' !"y #il = 1 – (36/52)(35/51)(34/50)(33/49) . $ its gr!tr th!" 0.5, t!E th g!'. )f the chance of you seein' a shootin' sta# outside -ithin one hou# is D! (e#cent, -hat is the chance of you seein' at least one shootin' sta# outside in 3" minutes? @h%%ti"g st!rs $%ll%w ! %iss%" istri&ti%" with #!r!'tr l!'!, !" "%t si"g ! sh%%ti"g st!r i" %" h%&r h!s #r%!ility 16J. 0.16 = #(Dl!'!), l!'! / 2 (c!&s %$ h!l$ !" h%&r) = D l" (0.4) = 0.92. @% 1 D #(D0.92) = 60J ch!"c %$ si"g ! sh%%ti"g st!r i" 30 'i"&ts.
•
•
•
•
•
•
•
•
•
•
•
)f ) #oll a dice and (ay you the amount sho-in', ho- much -ould you (ay to (lay? K %$ ic = L3.50 )f you a#e 'iven the choice to #e#oll afte# the initial #oll, main' the initial #esult null, and you a#e (aid based on the second #oll. o- much -ould you (ay to (lay? irst r%ll is w%rth L3.50, s% y%& w%&l %"ly r%ll sc%" i$ $irst r%ll w!s "%t ! 4, 5, %r 6. @% th sc%" r%ll is w%rth L5. 0.5 * 3.5 + 0.5 * 5 = L4.25 Th#ee #olls? irst r%ll is w%rth L3.50. @c%" r%ll is w%rth L5. irst tw% r%lls !r w%rth L4.25, s% y%& w%&l %"ly E# $irst r%ll i$ y%& gt ! 5 %r ! 6. 1/3 * 5.5 + 1/3 * 5 + 1/3 * 3.5 = L4.67 Eiven a cha#t -ith diffe#ent (oe# tou#naments -ith diffe#ent ent#y fees, #e-a#d (ots, and (#obabilities to -in, e(lain ho- you -ould 'au'e the #is a'ainst (otential (#ofits. Eiven a bunch of balls of diffe#ent -ei'hts and a scale, and all the balls loo identical but one -ei'hs diffe#ent, 'ive the minimum numbe# of t#ies -ith the scale to find the heavy ball. Fight h!l$ !t ! ti'. "swr is th #%wr %$ 2 th!t th "&'r %$ !lls is cl%sst t%. 8 family has 2 ids. )f you have seen 1 'i#l, -hat is the (#obability that the othe# id is a boy? $tr si"g %" girl, th #%ssil c%'i"!ti%"s !r girlD%y, %yDgirl, !" girlDgirl. hr$%r th #r%!ility th!t th %thr Ei is ! %y is 2/3. )f you fli( a coin until you decide to sto( and you -ant to maimi:e the #atio of heads to total fli(s, -hat is that e(ected #atio? 50J ch!"c %$ 1A1 r!ti% i"iti!lly ($li# h!s $irst), c!&s th" y%& w%&l st%#. $ y%& st!rt with ! t!il, y%& w%&l E# $li##i"g &"til y%& h! ! 1A2 r!ti% (50J h!s). hr$%r y%&r #ct r!ti% is 3A4. Find the heavie# ball of the 'iven 12 (ool balls usin' a -ei'ht balance that can be used ust 3 times. Fight 6 s. 6, th" 3 s. 3, th" 1 s. 1. T-o (eo(le tae tu#ns countin' to 0". $ach (e#son can add f#om 1 to 1" to the total sum. The fi#st to count the numbe# 0" -ins. What is you# st#ate'y? Go you -ant to 'o fi#st? G%& w!"t t% st!rt $irst !" hit MchcE#%i"tN "&'rs 6, 17, 28, !" 39.
6ou have five (i#ates, #aned f#om 0 to 1 in descendin' o#de#. The to( (i#ate has the #i'ht to (#o(ose ho- 1"" 'old coins should be divided amon' them .Hut the othe#s 'et to vote on his (lan, and if fe-e# than half a'#ee -ith him, he 'ets illed. o- -ill the coins end u( bein' divided, assumin' all the (i#ates a#e #ational and -ant to end u( alive? 6ou have D ma#bles and they all a#e identical looin' in si:e, colo#, and sha(e. Ine ma#ble -ei'hs mo#e than the othe# 7. 6ou have a balance and can use it t-ice. o- -ould you find out -hich ma#ble is the heavy one? Fight 3 !" 3. $ th s!', th" wight th %thr tw% t% $i" h!ir %". $ i$$r"t, t!E 2 %&t %$ th h!ir 3 !" wight th' t% $i" th h!ir %".
2Dic s&' #r%!ilitisA 2A 1 3A 2/36 = 1/18 4A 3/36 = 1/12 5A 4/36 = 1/9 6A 5/36 7A 6/36 = 1/6 8A 5/36
9A 4/36 = 1/9 10A 3/36 = 1/12 11A 2/36 = 1/18 12A 1/36 @q&!rs 1A 1 2A 4 3A 9 4A 16 5A 25 6A 36 7A 49 8A 64 9A 81 10A 100 11A 121 12A 144 13A 169 14A 196 15A 225 16A 256 17A 289 18A 324 19A 361 20A 400 >!ys th%r' (B>) = (>B) () / (>)