Proility Proility Curriulum Ry ACMSP:, 6, 6, 7 www.mthltis.om
Proility PROBABILITY Proility msurs th hn of somthing hppning. This mns w n us mthmtis to fin how likly it is tht n vnt will hppn. Answr ths qustions, for working through th hptr. I us to think: Wht is n outom? Wht is th rng of proility from impossil to rtin? Wht is th proility of rolling 6 with singl i? Answr ths qustions, ftr working through th hptr. But now I think: Wht is n outom? Wht is th rng of proility from impossil to rtin? Wht is th proility of rolling 6 with singl i? Wht o I know now tht I in't know for? 00% Proility Mthltis 00% P Lrning J
Proility Bsis Bsi Dfinitions Hr r four si trms for proility: Evnt An vnt is sitution whih oul hv iffrnt outoms. Outom An outom is possil rsult of n vnt. Smpl Sp All th possil outoms. For xmpl, th sm sp of rolling i is {,,,,, 6} n th smpl sp of tossing oin is {H, T}. Rnom Evnt An vnt with qully likly outoms. Lt's sy g ontins r stons n lu stons. On of th stons is pull out Th vnt is tht ston is slt from th g. Th possil outoms r tht r ston or lu ston oul slt. So th smpl sp is {R, B}. Th vnt is rnom vnt us thr r th sm numr of r stons s lu stons. If thr ws only on lu ston, thn th vnt woul not rnom us it is mor likly tht r ston is rwn. Rltiv Frquny Eh outom hs rltiv frquny whr Rltiv frquny numr oftims outom ours totlnumroftrils So rltiv frquny n hng from xprimnt to xprimnt. Hr is n xmpl: A g is full of 0 r, 0 lu n 0 grn stons. Lt's sy stons r rwn from th g t rnom in this orr: R G G B G G B B R R R G Th rltiv frquny of r stons is Th rltiv frquny of lu stons is Th rltiv frquny of grn stons is. or.. or Th nxt tim stons r rwn, thy r rwn in this orr: Th rltiv frquny of r stons is 6 Th rltiv frquny of lu stons is. B B G R R B R R B R B R G G or. 7 Th rltiv frquny of grn stons is. Rltiv frquny is lso somtims ll "xprimntl proility". J 00% Proility Mthltis 00% P Lrning
Proility Qustions Bsis. A oin is toss on. Wht is th smpl sp (th possil outoms)? Is this rnom vnt? Why?. A singl i is roll tn tims n ths r th numrs it shows:,,,,, 6,,,, Wht is th smpl sp of rolling i n is rolling i rnom vnt? Why? In this xprimnt, wht is th rlvnt frquny of ing roll? In this xprimnt, wht is th rlvnt frquny of ing roll?. A ox ftory noti tht 6 out of 7 oxs wr rokn. Wht is th rltiv frquny of rokn oxs? Wht is th rltiv frquny of unrokn oxs? If th ox ftory prou 600 oxs, wht numr of ths oxs is likly to rokn? 00% Proility Mthltis 00% P Lrning J
Proility Qustions Bsis. A fruit shop uys fruit oring to this tl: Fruit Numr ought Appls 8 Phs 78 Apriots 8 Orngs 7 Bnns 8 How mny itms of fruit wr ought in totl? Wht is th rltiv frquny of priots? Wht is th rltiv frquny of orngs? Th ownr notis tht of th ppls h ought wr rottn. Wht is th frquny of rottn ppls within th ppls? If th ownr ought 60 ppls ovr wk, how mny ppls shoul thy xpt to rottn? J 00% Proility Mthltis 00% P Lrning
Proility Knowing Mor Proility A g ontins r ston, lu ston n grn ston. If you hoos ston without looking, wht r th hns you will hoos r ston? Th proility woul. This is th formul to fin th proility of n outom "X": numrofwysx oul our P ^Xh totl numr ofoutoms Th 'totl numr of outoms' is lso th siz of th smpl sp. Hr is n xmpl: A singl i is roll, nswr th following qustions How mny possil outoms r thr? 6 (th i oul roll,,,, or 6) Wht is th proility tht is roll? numr of wys to roll P^h totlnumrofoutoms or 06. o 6 Wht is th proility tht n vn numr is roll? P^vnh numr of vn rolls totlnumrofoutoms or 0. Wht is th proility tht numr grtr thn is roll? numr of rollsgrtr thn P^ h totl numr ofoutoms or 06. o 6 6 Th formul for proility oul lso us in othr wys. A wthmkr slts 00 of his wths t rnom n notis 6 of thm r fulty. Wht is th rltiv frquny of fulty wths? numroffultywths 6 00 00 If th wthmkr m 00 wths in totl, how mny woul h xpt r fulty? P^fultyh numroffultywths totlnumrofwths 6 numroffultywths 00 00 numr of fulty wths 00 # 6 90 00 This will not finitly hppn. It is just n xpt vlu s on th proility. 00% Proility Mthltis 00% P Lrning J
Proility Knowing Mor Proility is Alwys Btwn 0 n Th numr of wys ny outom X n our will lwys lss thn or qul to th totl numr of outoms. So th proility will lwys frtion. This mns tht: 0 # P^Xh# If P ^X h 0 it mns tht th outom X is impossil. If P ^X h, it mns tht th outom X is rtin. Th losr P^Xh is to 0, th mor unlikly X is n th losr P^Xh is to th mor likly X is. If you fin proility grtr thn thn mistk hs n m somwhr. Complmntry Proility Th omplmnt of n outom is whn th outom osn't our. Th nottion X u (u is ll til) mns th omplmnt of X n so P ^ X u h mns th proility of X not ouring. Th totl proilitis of th outoms must up to n so P^Xh + P^Xu h. This mns w n us th formul: P^Xu h -P^Xh A g ontins lu stons, grn stons n r stons. Answr ths qustions if on ston is rwn t rnom Fin th proility tht th ston will not lu. Fin th proility tht th ston will not r. P^not luh -P^luh - P^not rh -P^rh - 7 Fin th proility tht th ston will not grn. P^not grnh -P^grnh - 6 J 00% Proility Mthltis 00% P Lrning
Proility Qustions Knowing Mor. A ook hs 0 pgs n is opn to rnom pg. Wht is th proility it opns on pg 89? Wht is th proility it is opn to n o pg? Wht is th proility of opning to pg 6 or ftr? Wht is th proility of opning to pg ftr pg 6? Wht is th proility tht it is not opn to pg 0? f Wht is th proility tht it is opn to pg numr whih is multipl of? g Wht is th proility tht th pg is not multipl of? h How r th nswrs from f n g rlt? Why? 00% Proility Mthltis 00% P Lrning J 7
Proility Qustions Knowing Mor. You noti tht whn tking tst, you gt out of th first qustions inorrt. Wht is th xprimntl proility of inorrt nswrs s on this informtion? Bs on this informtion, how mny qustions woul you xpt r inorrt if th tst totll 80 qustions?. A stnr k of rs (with no jokrs) is shuffl n pl f own n spr out. If r is rwn t rnom, wht is th siz of th smpl sp? If r is rwn t rnom, wht is th proility it is n? If r is rwn t rnom, wht is th proility it is th of sps? If r is rwn t rnom, wht is th proility it is not imon? If r is rwn t rnom, wht is th proility it is r? 8 J 00% Proility Mthltis 00% P Lrning
Proility Qustions Knowing Mor. A lirry hs ooks out sport, foo n history only. Th proility of slting singl history ook is 9. How mny history ooks woul you xpt to in group of 6 rnomly slt ook? Th proility of slting singl ook out foo is 9. Wht is th proility of slting singl ook out sport? If ooks r slt t rnom, how mny of h typ of ook oul xpt to slt? 00% Proility Mthltis 00% P Lrning J 9
Proility Qustions Knowing Mor. Th lttrs from th wor Mississippi r put into g. Answr th following qustions if singl lttr is rwn: Wht is th smpl sp of this xprimnt? (Hint: How mny iffrnt lttrs r thr?) Whih lttrs hv th grtst hn of ing slt? Why? Wht is th proility of slting n 's'? Wht is th proility of not slting th 'M'? 0 J 00% Proility Mthltis 00% P Lrning
Proility Using Our Knowlg Mutully Exlusiv Evnts Mutully xlusiv vnts r vnts tht nnot our t th sm tim. If thy r not mutully xlusiv thy r ll inlusiv vnts. For xmpl, rolling n rolling with singl i r mutully xlusiv vnts thy n't hppn t th sm tim. If X n Y r mutully xlusiv vnts thn P^Xor Yh P^Xh+ P^Yh. Fin th proility of rolling or with singl i P^or h P^h+ P^h + 6 6 Hr is n xmpl ompring mututlly xlusiv vnts n inlusiv vnts: A ook hs 0 pgs n is opn to rnom pg. Whih of ths r mutully xlusiv? X: Th pg numr is multipl of X: Th pg numr is multipl of 0 Y: Th pg numr is vn Y: Th pg numr is o Ths r inlusiv vnts us 0 n 0 r multipls of n vn whih mns X n Y n hppn t th sm. Ths r mutully xlusiv vnts us thr r no numrs whih r multipls of 0 n o. So X n Y n't hppn t th sm tim. ` P^Xor Yh! P^Xh+ P^Yh ` P^Xor Yh P^Xh+ P^Yh + 0 0 0 06. Numrs from 0 Numrs from 0 Multipls of 7 0 0 9 9 7 8 6 6 8 Evn numrs O Numrs 9 7 7 9 0 0 8 6 8 6 Multipls of 0 In th Vnn igrm ov it's sy to s tht 0 n 0 r oth multipls of AND vn numrs. Th sts ovrlp. In th Vnn igrm ov it's sy to s tht th sts of multipls of 0 n o numrs r mutully xlusiv. Th sts o not ovrlp n r sprt. 00% Proility Mthltis 00% P Lrning J
Proility Using Our Knowlg If X n Y r inlusiv thn thr is n xtr stp to fin P^Xor Yh. Sutrt th proility of th 'ovrlpping' outoms. To ontinu th xmpl from th prvious pg. A ook hs 0 pgs n is opn to rnom pg. Fin th proility tht th pg numr is vn OR multipl of X: Th pg numr is multipl of Numrs from 0 Y: Th pg numr is vn P^Xor Yh P^Xh+ P^Yh- P^Xn Yh + 0-0 0 0 06. 0 Sin som pg numrs r in oth X n Y thy shouln't ount twi Multipls of 9 7 0 9 7 0 8 6 6 8 Evn numrs Compoun Evnts A ompoun vnt involvs mor thn on outom. It oul hv two stgs or mor. To fin th proility of ompoun vnts, fin th proility of h outom n multiply thm togthr. A g hols 6 r stons n lu stons. Fin th proility of rwing two lu stons from two rws. Stp : Fin th proility tht th first ston is lu. P^ststonisluh numroflustons totlstons 0 Stp : Fin th proility tht th son ston is lu. numroflustonsrmining P^nstonisluh totlstonsrmining 9 Thr is on lss lu ston from th prvious rw Thr is on lss ston in th g from th prvious Stp : Multiply th proilitis togthr: P^lustonsh P^ststonis luh# P^n stonisluh # J 00% Proility Mthltis 00% P Lrning
Proility Using Our Knowlg Tr Digrms Tr igrms r us with ompoun vnts to s ll th possil outoms (th smpl sp). A g ontins r, grn n lu ston. Wht r ll th possil mthos to slt two stons? First rw Son rw Smpl sp R B G R B G R B G R B R R R B B B G G R B G R B G R B Smpl sp (ll possil outoms) From th tr igrm w n tht thr r 8 possil outoms in this smpl sp. {RR, RB, RG, BR, BB, BG, GR, GB} (GG is not in th smpl sp us thr is grn ston.) Wht is th proility tht lu ston is slt first n r ston slt son? (without rpling th lu ston) P^BRh # Wht is th proility of rwing on r n on grn ston? Thr r two possil outoms with on r ston n on grn tl: GR n RG P^GR or RGh P^GRh+ P^RGh ` # # j+ ` j 0 Noti th iffrn twn n. In, th orr mttrs so thr is only on wy to rw lu first n r son. In, th orr osn't mttr, so oth GR n RG r ount. 00% Proility Mthltis 00% P Lrning J
Proility Using Our Knowlg Tls for Two-Stg Evnts If th ompoun vnt is just two-stg vnt, thn two-wy tl n us. Two multipl hoi qustions with options A, B n C n to nswr. How mny possil wys r thr to nswr th qustions? Q Q A B C A AA BA CA B AB BB CB C AC BC CC Thr r 9 possil wys to nswr th Q n Q: {AA, BA, CA, AB, BB, CB, AC, BC, CC}. So th smpl sp siz is 9. Wht is th proility tht oth nswrs r A? Only on outom hs oth nswrs s A. ` P^oth nswrsr Ah P^AAh 9 Wht is th proility tht oth nswrs r th sm? P^sm nswrsh P^AAor BBor CCh P^AAh+ P^BBh+ P^CCh + + 9 9 9 Wht r th hn tht th nswrs r iffrnt? P^iffrnt nswrsh -P^sm nswrsh - To fin th smpl siz of ompoun vnt without tl or tr igrm, multiply th smpl sizs of h stg. In th xmpl ov thr r wys to nswr Q n wys to nswr Q, so th smpl siz is # 9. A rsturnt srvs strtrs, mins n ssrts. How mny wys r thr to orr thr ours ml of strtr, min, n ssrt? strtrs # mins# ssrts # # 60 iffrntwystoorrthroursml. J 00% Proility Mthltis 00% P Lrning
Proility Qustions Using Our Knowlg. Wht is th iffrn twn mutully xlusiv vnts n inlusiv vnts?. Intify in th following if outoms A n B r mutully xlusiv or not. Giv rson why you sy so. A: Otining 'hs' from oin toss A: Finishing tsk twn Mony n Thursy B: Otining 'tils' from oin toss B: Finishing tsk twn Stury n Tusy. A singl i is roll. Answr th qustions out ths outoms (hk if thy r mutully xlusiv first): A: Rolling or 6 B: Rolling n vn numr C: Rolling n o numr D: Rolling Fin P (B or D). Fin P (A or D). Fin P (A or C). Fin P (B or C). 00% Proility Mthltis 00% P Lrning J
Proility Qustions Using Our Knowlg. Us this informtion to nswr th following qustions: P ^Ah P ^Bh 0 Ar A n B mutully xlusiv? P ^Ch P ^Dh P ^Aor Bh Ar C n D mutully xlusiv? P ^Cor Dh 7 00 P ^Bor Ch Ar B n C mutully xlusiv? If P (A or C), us P (A or C) P (A) + P (C) - P (A n C) to fin P (A n C). Fin P (B n C). 6 J 00% Proility Mthltis 00% P Lrning
Proility Qustions Using Our Knowlg. A ook with 0 pgs is opn up to rnom pg. Answr qustions tht follow out ths outoms: (Hint: First hk if thy r mutully xlusiv) A: Th pg numr is multipl of 0 B: Th pg numr hs 7 in it C: Th pg numr is multipl of D: Th pg numr is multipl of E: Th pg numr is 0 or low F: Th pg numr is or mor Fin P (A or B). Fin P (A or F). Fin P (B or E). Fin P (C or D). Fin P (E or F). f Fin P (B or D). 00% Proility Mthltis 00% P Lrning J 7
Proility Qustions Using Our Knowlg 6. A singl i is roll twi. Wht is th proility of rolling n thn? Wht is th proility tht oth rolls will grtr thn? Wht is th proility tht oth rolls will vn? Wht is th proility of oth rolls ing 6? Wht is th proility tht oth rolls will or? 8 J 00% Proility Mthltis 00% P Lrning
Proility Qustions Using Our Knowlg 7. A g ontins yllow, whit n orng ston. A ston is rwn t rnom n thn rpl. Thn ston is rwn t rnom for son tim. Complt th tr igrm low for this ompoun vnt: First rw Son rw Smpl sp Y Y - Y Y Y - W O Y - - W - W - O - O - - How ig is th smpl sp of this xprimnt? Is this wht you xpt? Wht is th proility th whit ston will rwn first? Wht is th proility th whit ston will rwn son? Wht is th proility tht th whit ston will rwn oth tims? 00% Proility Mthltis 00% P Lrning J 9
Proility Qustions Using Our Knowlg f Rrw th tr igrm if th ston tht is rwn first is not rpl? g Wht is th siz of th smpl sp now? h Wht is th proility tht th yllow ston will rwn first? i Wht is th proility tht th yllow ston will rwn son? j Wht is th proility tht th yllow ston is rwn oth tims? 0 J 00% Proility Mthltis 00% P Lrning
Proility Qustions Using Our Knowlg 8. A tnnis tournmnt hs singls trophy n ouls trophy. Th ountris ompting for th singls trophy r: Ini, Spin n Gr. Th ountris ompting for th ouls trophy r just Ini n Spin. Eh ountry hs qul hn to win th trophis. Drw tl for this two-stg vnt of trophy winnrs. Wht is th proility tht Gr will win th singls trophy? Wht is th proility tht Ini will win oth trophis? Wht is th proility tht Ini n Spin will win trophy h? Wht is th proility tht Spin n Gr will win trophy h? f Wht is th proility tht Ini will not win trophy? 00% Proility Mthltis 00% P Lrning J
Proility Thinking Mor Proility Tr Digrms Whn proilitis of h outom r writtn on th 'rnhs' of tr igrm it is ll proility tr igrm. To fin th proility of h multi-stg vnt just multiply th proilitis of h rnh. Th proility of flipping hs with trik oin is Drw tr igrm rprsnting flips of this oin: # # H T # H T H # T H T H T H T H T H H H H H T H T H H T T T H H T H T T T H T T T Fin th proility of flipping hs? P^HHHh # # 6 Multiply th proilitis of h rnh Fin th proility of flipping T-H-T. P^THTh # # Multiply th proilitis of h rnh J 00% Proility Mthltis 00% P Lrning
Proility Thinking Mor Proility from Vnn Digrms Vnn igrms hlp us s groups ttr. Lt's sy tht from 0 stunts, ply footll, ply tnnis n ply oth thn th Vnn igrm woul look lik this: Footll Tnnis 7 6 So th igrm shows tht 7 stunts ply footll only, 6 stunts ply tnnis only, stunts ply oth n stunts on't ply ithr gm. Th prt in oth irls is ll th intrstion. It hs this symol "+". In th xmpl ov thr r (7 + ) stunts in footll, (6 + ) in tnnis n in th intrstion footll + tnnis. This mns tht stunts ply footll AND tnnis. Th union of sts is th omin st n hs th symol ",". Aov xmpl footll, tnnis hs 7 + + 6 8 stunts. This mns tht stunts ply footll OR tnnis. Lt's sy stunt is slt t rnom from th group rprsnt in th ov Vnn igrm Fin th proility tht th stunt plys footll. P^footllh numroffootllplyrs numrs of stunts 7 0 + Fin th proility tht th stunt plys tnnis or footll. P^footll numrofstuntswho ply footll or tnnis, tnnish numrsofstunts 7 + + 6 0 Fin th proility tht th stunt plys tnnis n footll. numrofstuntswho ply footll ntnnis P^footll + tnnish numrs of stunts 0 6 00% Proility Mthltis 00% P Lrning J
Proility Thinking Mor Proility from Tls Somtims tls r us in proility qustions. At th olympis, ountry won mls for vnts oring to this tl Gol Silvr Bronz Totl Swimming 7 Fning 0 8 Arhry 6 9 Totl 7 9 00 If ml is slt t rnom thn wht is th proility it is gol? numrofgolmls P^gol mlh totlmls 7 00 If silvr ml is slt t rnom thn wht is th proility it woul for rhry? numr of silvrmlforrhry P^silvr ml forrhryh totlsilvrmls 9 If ml from fning is slt t rnom, wht is th proility it is ronz? numr of ronz fning mls P^fning ml is ronzh totl fning mls 8 If ml is slt t rnom, wht is th proility it will gol ml for swimming? numrofgolmls for swimming P^gol mlfrom swimmingh totlmls 00 Th typ of tl in th ov xmpl is ll 'ontingny' tl. J 00% Proility Mthltis 00% P Lrning
Proility Qustions Thinking Mor. Wht o th symols + n, mn?. A group of 0 popl r sk whih pts thy hv. This is rprsnt in this Vnn Digrm: Cts Dogs 7 9 How mny popl hv ogs only? How mny popl hv ogs n ts? If prson is hosn t rnom fin th proility tht thy hv ts only, P (ts). Fin th proility of prson hving ts n ogs, P (ts + ogs). Fin th P (ts, ogs). 00% Proility Mthltis 00% P Lrning J
Proility Qustions Thinking Mor. A iffrnt group of popl ws sk out whih pts thy h n this is th rsulting Vnn igrm: Cts 6 7 Dogs 6 0 Fish How mny popl hv fish only? How mny r in th st ts + ogs? How mny popl in th st ogs + fish? How mny popl hv ll pts? If prson is slt t rnom, thn fin P (ts + ogs). f If prson is slt t rnom, thn fin P (ts, ogs). g If prson is slt t rnom, thn fin P (ogs + fish). 6 J 00% Proility Mthltis 00% P Lrning
Proility Qustions Thinking Mor. A prolity tr igrm for rtin ompoun vnt looks lik this: A 6 0 C D 0 7 0 I J K L M N A C I A C J A C K A D L A D M A D N E 8 8 O P Q A E O A E P A E Q B 0 0 F G H 9 0 0 0 9 0 R S T U V W X Y Z B F R B F S B F T B G U B G V B G W B H X B H Y B H Z How mny stgs r in this ompoun vnt? Us th tr igrm to fin P (AD). Us th tr igrm to fin P (BGU). Us th tr igrm to fin P (AEQ). Us th tr igrm to fin P (ADN). 00% Proility Mthltis 00% P Lrning J 7
Proility Qustions Thinking Mor. Th tl low is of group of tngrs who ply iffrnt sports. Tnnis Footll Vollyll Totl Boys 8 Girls 9 6 6 Totl 0 6 0 If stunt is slt t rnom wht is th proility of slting r oy? If stunt is slt t rnom wht is th proility of slting vollyll plyr? If stunt is slt t rnom wht is th proility of slting girl who plys footll? If oy is slt t rnom wht is th proiliy h plys footll? If footll plyr is slt t rnom, wht is th proility thy r girl? 8 J 00% Proility Mthltis 00% P Lrning
Proility Answrs Bsis: Knowing Mor:. Hs n tils or {H,T}. This is rnom vnt us h outom is qully likly.. Th siz of th smpl sp is.. Th smpl sp of rolling i is {,,,,, 6} n it is rnom vnt us h outom is qully likly.. Rltiv frquny of ing roll is 0. Rltiv frquny of ing roll is 0. Rltiv frquny of rokn oxs is.. 8 Foo: History: Sport: 8 9 Rltiv frquny. {m, i, s, p} 00 'i'n 's' s thy ppr tims h in th wor Mississippi.. 00 Proility of slting n 's' is. Rltiv frquny of priots is. 00 0 Rltiv frquny of orngs is 6. Rltiv frquny of rottn ppls within th ppls is. 7. Using Our Knowlg: Mutully xlusiv vnts nnot our t th sm tim. Appls xpt to rottn... Knowing Mor: Mutully inlusiv vnts n our t th sm tim.. g h 0 9 0 7 Whn you th two proilitis, thy qul. This is us ths r ll th possil outoms, n ll th possil outoms hv proility of us proility of ovrs ll outoms. f. Rolling 'hs' n 'tils' in oin toss r mutully xlusiv vnts. This is us ths nnot on t th sm tim. Finishing tsk twn Mony n Thursy n finishing tsk twn Stury n Tusy r mutully inlusiv vnts. This is us thy n on t th sm tim. Th tsk oul omplt on Mony or Tusy. 00% Proility Mthltis 00% P Lrning J 9
Proility Answrs Using Our Knowlg:. P (B or D) 7. P (A or D) P (A or C) P (B or C) Using Our Knowlg: First rw Son rw Smpl sp Y Y - Y Y W Y - W O Y - O W Y W W - Y W - W. Ys us P^Ah+ P^Bh P^Aor Bh O Y W - O - O Y No us P^Ch+ P^Dh! P^Cor Dh O W O - W No us P^Bh+ P^Ch! P^Bor Ch O O - O P (A n C) 0 P (B n C) 0 Th smpl sp hs 9 possil outoms. This is xpt sin outoms r in h vnt. So, thr shll # 9 totl outoms.. P^Aor Bh P^Aor Fh P^Bor Eh 0 f 9 Y W Y - W P^Cor Dh O Y Y - O W - Y P^Eor Fh W f P^Bor Dh O Y W - O O - Y O 6. 6 9 W O - W 9 6 g h j Thr r now 6 totl outoms. 0 i 0 J 00% Proility Mthltis 00% P Lrning
Proility Answrs Using Our Knowlg: 8.. Singls Trophy. f Douls Trophy Ini Spin Thinking Mor: Ini Spin Gr Ini, Ini Ini, Spin Spin, Ini Spin, Spin 6 6 Gr, Ini Gr, Spin + is mthmtil symol for th trm intrstion. For xmpl, A+ B is st whih ontins ll th lmnts tht sts A n B hv in ommon.. Thinking Mor: Thr r stgs. 9 P^oyh P^vollyllh 7 P^girl plys footllh P^oy plys footllh 00 0 P^footll n girlh 6, is mthmtil symol for th trm union. For xmpl, A, B is st whih ontins ll th lmnts tht r in A or in B or in oth A n B.. 9 P^tsh 7 0 P^ts + ogsh P ^ts, ogsh. 9 6 9 0 f 7 8 g 0 00% Proility Mthltis 00% P Lrning J
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