{"text_format":"1","workbook":"xlsx","n_sheets":4,"sheets":["Birthday calculations","Birthday simulation","Red bag, blue bag","Let's make a deal!"],"not_rendered":{"charts":1,"images":0,"macros":1,"external_links":1,"defined_names":10,"comments":0,"pivot_tables":0},"unopened":[],"bounds":{"document_bytes":16777216,"sheet_cells":100000,"rows_first":2000,"rows_last":500,"rows_between":2000,"halvings":0,"cell_chars":16384,"cell_head_chars":4096},"cells_cut":0,"cells_raw":0,"read":"whole","date_system":"1900"}

{"sheet":"Birthday calculations","index":0,"n_rows":66,"n_cols":14,"rows_kept":66,"cut":false,"hidden":false,"merged":[[0,1,0,6]]}
[null,"Birthdays (among 60 people)"]
[]
[null,null,"Pr(born on different date than predecessors | predecessors all born on different dates)"]
[null,null,null,"Pr(all up to this person born on distinct dates)"]
[null,null,null,null,"Pr(by the time we reach this person, at least two with same birthdate)"]
[null,"person",null,null,null,"the same, computed directly"]
[null,1,1,{"$f":"PRODUCT($C$7:C7)","v":1},{"$f":"1-D7","v":0},0]
[null,2,{"$f":"365/366","v":0.99726775956284153},{"$f":"PRODUCT($C$7:C8)","v":0.99726775956284153},2.732240437158473E-3,{"$f":"1*F7+(B7/366)*(1-F7)","v":2.7322404371584699E-3}]
[null,3,{"$f":"364/366","v":0.99453551912568305},{"$f":"PRODUCT($C$7:C9)","v":0.99181820896413742},8.1817910358625845E-3,{"$f":"1*F8+(B8/366)*(1-F8)","v":8.181791035862522E-3}]
[null,4,{"$f":"(366-B9)/366","v":0.99180327868852458},{"$f":"PRODUCT($C$7:C10)","v":0.98368855151361168},1.6311448486388325E-2,1.6311448486388241E-2]
[null,5,{"$f":"(366-B10)/366","v":0.98907103825136611},{"$f":"PRODUCT($C$7:C11)","v":0.97293785696155033},2.7062143038449671E-2,2.7062143038449574E-2]
[null,6,0.98633879781420764,{"$f":"PRODUCT($C$7:C12)","v":0.95964635618338701},4.0353643816612994E-2,4.0353643816612828E-2]
[null,7,0.98360655737704916,{"$f":"PRODUCT($C$7:C13)","v":0.94391444870497077},5.6085551295029235E-2,5.6085551295029012E-2]
[null,8,0.98087431693989069,{"$f":"PRODUCT($C$7:C14)","v":0.92586144012318172},7.4138559876818277E-2,7.4138559876818083E-2]
[null,9,0.97814207650273222,{"$f":"PRODUCT($C$7:C15)","v":0.90562403159589899},9.4375968404101007E-2,9.4375968404100744E-2]
[null,10,0.97540983606557374,{"$f":"PRODUCT($C$7:C16)","v":0.88335458819599977},0.11664541180400023,0.1166454118039999]
[null,11,0.97267759562841527,{"$f":"PRODUCT($C$7:C17)","v":0.85921921693381398},0.14078078306618602,0.14078078306618569]
[null,12,0.9699453551912568,{"$f":"PRODUCT($C$7:C18)","v":0.83339568855602175},0.16660431144397825,0.16660431144397791]
[null,13,0.96721311475409832,{"$f":"PRODUCT($C$7:C19)","v":0.80607123975090622},0.19392876024909378,0.19392876024909339]
[null,14,0.96448087431693985,{"$f":"PRODUCT($C$7:C20)","v":0.77744029407669368},0.22255970592330632,0.22255970592330593]
[null,15,0.96174863387978138,{"$f":"PRODUCT($C$7:C21)","v":0.74770214075135566},0.25229785924864434,0.25229785924864395]
[null,16,0.95901639344262291,{"$f":"PRODUCT($C$7:C22)","v":0.71705861039269347},0.28294138960730653,0.28294138960730608]
[null,17,0.95628415300546443,{"$f":"PRODUCT($C$7:C23)","v":0.6857117858946522},0.3142882141053478,0.31428821410534735]
[null,18,0.95355191256830596,{"$f":"PRODUCT($C$7:C24)","v":0.65386178491047431},0.34613821508952569,0.34613821508952519]
[null,19,0.95081967213114749,{"$f":"PRODUCT($C$7:C25)","v":0.62170464794766411},0.37829535205233589,0.37829535205233544]
[null,20,0.94808743169398912,{"$f":"PRODUCT($C$7:C26)","v":0.58943036294491657},0.41056963705508343,0.41056963705508304]
[null,21,0.94535519125683065,{"$f":"PRODUCT($C$7:C27)","v":0.55722105349437467},0.44277894650562533,0.44277894650562494]
[null,22,0.94262295081967218,{"$f":"PRODUCT($C$7:C28)","v":0.52524935370371384},0.47475064629628616,0.47475064629628583]
[null,23,0.93989071038251371,{"$f":"PRODUCT($C$7:C29)","v":0.49367698818053979},0.50632301181946016,0.50632301181945993,null,{"$f":"(365/366)^59","v":0.85093221297193167},"Pr(a particular person has a birthdate all to him/herself)"]
[null,24,0.93715846994535523,{"$f":"PRODUCT($C$7:C30)","v":0.46265357089050591},0.53734642910949404,0.53734642910949382,null,{"$f":"showformula(H29)","v":"=(365/366)^59"}]
[null,25,0.93442622950819676,{"$f":"PRODUCT($C$7:C31)","v":0.43231563181571864},0.56768436818428136,0.56768436818428114]
[null,26,0.93169398907103829,{"$f":"PRODUCT($C$7:C32)","v":0.40278587554415318},0.59721412445584687,0.59721412445584665,null,{"$f":"showformula(C46)","v":"=(366-B45)/366"},null,{"$f":"showformula(D46)","v":"=PRODUCT($C$7:C46)"},null,null,{"$f":"showformula(E46)","v":"=1-D46"},{"$f":"showformula(F46)","v":"=1*F45+(B45/366)*(1-F45)"}]
[null,27,0.92896174863387981,{"$f":"PRODUCT($C$7:C33)","v":0.37417267127052484},0.62582732872947511,0.625827328729475]
[null,28,0.92622950819672134,{"$f":"PRODUCT($C$7:C34)","v":0.3465697692915517},0.6534302307084483,0.65343023070844819,null,"The final column is based on:"]
[null,29,0.92349726775956287,{"$f":"PRODUCT($C$7:C35)","v":0.32005623502881003},0.67994376497118991,0.67994376497118991]
[null,30,0.92076502732240439,{"$f":"PRODUCT($C$7:C36)","v":0.29469658799100817},0.70530341200899183,0.70530341200899183,null,"Pr(among k people, at least two share a birthdate)"]
[null,31,0.91803278688524592,{"$f":"PRODUCT($C$7:C37)","v":0.27054112995895835},0.7294588700410416,0.72945887004104171]
[null,32,0.91530054644808745,{"$f":"PRODUCT($C$7:C38)","v":0.24762644408811763},0.75237355591188237,0.75237355591188249,null,"= Pr(among k, sharing occurs | sharing occurs amongst the first k-1)"]
[null,33,0.91256830601092898,{"$f":"PRODUCT($C$7:C39)","v":0.22597604460500351},{"$f":"1-D39","v":0.77402395539499647},0.77402395539499658,null,"   ∙ Pr(sharing occurs amongst the first k-1)"]
[null,34,0.9098360655737705,{"$f":"PRODUCT($C$7:C40)","v":0.20560115533733925},0.79439884466266075,0.79439884466266086,null,"+ Pr(among k, sharing occurs | sharing doesn't occur amongst the first k-1)"]
[null,35,0.90710382513661203,{"$f":"PRODUCT($C$7:C41)","v":0.18650159445900719},0.81349840554099284,0.81349840554099295,null,"   ∙ Pr(sharing doesn't occur amongst the first k-1)"]
[null,36,0.90437158469945356,{"$f":"PRODUCT($C$7:C42)","v":0.16866674252986716},0.83133325747013287,0.83133325747013298]
[null,37,0.90163934426229508,{"$f":"PRODUCT($C$7:C43)","v":0.15207657113348677},0.84792342886651317,0.8479234288665134,null,"= 1 ∙ Pr(sharing occurs amongst the first k-1)"]
[null,38,0.89890710382513661,{"$f":"PRODUCT($C$7:C44)","v":0.13670271011725996},0.86329728988274002,0.86329728988274024,null,"+ (k-1)/366 ∙ [1 - Pr(sharing occurs amongst the first k-1)]"]
[null,39,0.89617486338797814,{"$f":"PRODUCT($C$7:C45)","v":0.12250953256410183},0.8774904674358982,0.87749046743589831]
[null,40,0.89344262295081966,{"$f":"PRODUCT($C$7:C46)","v":0.10945523811055},0.89054476188944998,0.89054476188945009]
[null,41,0.89071038251366119,{"$f":"PRODUCT($C$7:C47)","v":9.7492917005571847E-2},0.90250708299442817,0.90250708299442817]
[null,42,0.88797814207650272,{"$f":"PRODUCT($C$7:C48)","v":8.6571579308226371E-2},0.91342842069177366,0.91342842069177366]
[null,43,0.88524590163934425,{"$f":"PRODUCT($C$7:C49)","v":7.6637135781052856E-2},0.92336286421894709,0.9233628642189472]
[null,44,0.88251366120218577,{"$f":"PRODUCT($C$7:C50)","v":6.7633319282185994E-2},0.93236668071781403,0.93236668071781403]
[null,45,0.8797814207650273,{"$f":"PRODUCT($C$7:C51)","v":5.9502537729136308E-2},0.94049746227086373,0.94049746227086373]
[null,46,0.87704918032786883,{"$f":"PRODUCT($C$7:C52)","v":5.2186651942767087E-2},0.94781334805723294,0.94781334805723294]
[null,47,0.87431693989071035,{"$f":"PRODUCT($C$7:C53)","v":4.5627673829741713E-2},0.95437232617025824,0.95437232617025836]
[null,48,0.87158469945355188,{"$f":"PRODUCT($C$7:C54)","v":3.9768382381660125E-2},0.96023161761833986,0.96023161761833997]
[null,49,0.86885245901639341,{"$f":"PRODUCT($C$7:C55)","v":3.4552856823409618E-2},0.96544714317659042,0.96544714317659042]
[null,50,0.86612021857923494,{"$f":"PRODUCT($C$7:C56)","v":2.9926927904428548E-2},0.97007307209557148,0.97007307209557148]
[null,51,0.86338797814207646,{"$f":"PRODUCT($C$7:C57)","v":2.5838549775408254E-2},0.97416145022459177,0.97416145022459177]
[null,52,0.86065573770491799,{"$f":"PRODUCT($C$7:C58)","v":2.2238096118179233E-2},0.97776190388182072,0.97776190388182083]
[null,53,0.85792349726775952,{"$f":"PRODUCT($C$7:C59)","v":1.9078585194284913E-2},0.98092141480571504,0.98092141480571515]
[null,54,0.85519125683060104,{"$f":"PRODUCT($C$7:C60)","v":1.6315839250850211E-2},0.98368416074914977,0.98368416074914988]
[null,55,0.85245901639344257,{"$f":"PRODUCT($C$7:C61)","v":1.3908584279413293E-2},0.98609141572058667,0.98609141572058678]
[null,56,0.84972677595628421,{"$f":"PRODUCT($C$7:C62)","v":1.1818496477862116E-2},0.98818150352213785,0.98818150352213796]
[null,57,0.84699453551912574,{"$f":"PRODUCT($C$7:C63)","v":1.0010201934801247E-2},0.98998979806519871,0.98998979806519882]
[null,58,0.84426229508196726,{"$f":"PRODUCT($C$7:C64)","v":8.4512360597092506E-3},0.99154876394029079,0.99154876394029079]
[null,59,0.84153005464480879,{"$f":"PRODUCT($C$7:C65)","v":7.1119691431433037E-3},0.99288803085685673,0.99288803085685673]
[null,60,0.83879781420765032,{"$f":"PRODUCT($C$7:C66)","v":5.9655041719808592E-3},0.99403449582801917,0.99403449582801917]

{"sheet":"Birthday simulation","index":1,"n_rows":64,"n_cols":14,"rows_kept":64,"cut":false,"hidden":false,"merged":[[4,6,4,12],[5,6,5,12],[4,5,6,5],[0,1,0,5]]}
[null,"Birthdays (among 60 people)"]
[]
[null,"person","birthdate","total with this birthdate"]
[null,1,{"$f":"DATE(2008,1,1)+RANDBETWEEN(0,365)","v":"2008-04-14"},{"$f":"COUNTIF($C$4:$C$63,C4)","v":1}]
[null,2,"2008-02-17",1,null,"all distinct dates?","Simulation"]
[null,3,"2008-03-25",1,null,null,"number of people born on k-person date"]
[null,4,"2008-06-14",1,null,null,1,2,3,4,5,6,"more","k"]
[null,5,"2008-08-26",1,null,{"$f":"IF(G8=60,1,0)","v":1},{"$f":"COUNTIF($D$4:$D$63,G7)","v":60},0,0,0,0,0,{"$f":"60-SUM(G8:L8)","v":0},"people"]
[null,6,"2008-11-07",1]
[null,7,"2008-12-09",1,null,{"$f":"ADDRESS(ROW($F$8),COLUMN($F$8))","v":"$F$8"},{"$f":"ADDRESS(ROW($G$8),COLUMN($G$8))","v":"$G$8"},{"$f":"ADDRESS(ROW($H$8),COLUMN($H$8))","v":"$H$8"},{"$f":"ADDRESS(ROW($I$8),COLUMN($I$8))","v":"$I$8"},{"$f":"ADDRESS(ROW($J$8),COLUMN($J$8))","v":"$J$8"},{"$f":"ADDRESS(ROW($K$8),COLUMN($K$8))","v":"$K$8"},{"$f":"ADDRESS(ROW($L$8),COLUMN($L$8))","v":"$L$8"},{"$f":"ADDRESS(ROW($M$8),COLUMN($M$8))","v":"$M$8"},"monitored cell"]
[null,8,"2008-07-25",1,null,5.9199999999999999E-3,51.026820000000001,8.2861399999999996,0.65117999999999998,3.4079999999999999E-2,1.6000000000000001E-3,1.8000000000000001E-4,0,"mean"]
[null,9,"2008-07-27",1,null,7.6713834802625405E-2,3.6798119650005185,3.5429436101770304,1.3752539841190092,0.36894432479596156,8.9428854265274763E-2,3.286302481184588E-2,0,"sample standard deviation"]
[null,10,"2008-09-13",1,null,0,31,0,0,0,0,0,0,"minimum"]
[null,11,"2008-05-07",1,null,1,60,26,9,8,5,6,0,"maximum"]
[null,12,"2008-03-19",1,null,100000,100000,100000,100000,100000,100000,100000,100000,"number of simulation runs"]
[null,13,"2008-04-21",1]
[null,14,"2008-05-30",1]
[null,15,"2008-06-26",1,null,"Here, we see uses of the COUNTIF function: =COUNTIF(range, criterion).",null,null,null,null,null,null,{"$f":"showformula(G8)","v":"=COUNTIF($D$4:$D$63,G7)"}]
[null,16,"2008-04-02",1,null,"If the criterion is just a number, the function yields a count of the cells in the",null,null,null,null,null,null,"(count people born on k-person dates)"]
[null,17,"2008-04-05",1,null,"range which take that value."]
[null,18,"2008-08-08",1,null,null,null,null,null,null,null,null,{"$f":"showformula(D4)","v":"=COUNTIF($C$4:$C$63,C4)"}]
[null,19,"2008-01-05",1,null,"COUNTIF can also take either of two types of text criteria. Such criteria",null,null,null,null,null,null,"(count people with the same birthday)"]
[null,20,"2008-07-03",1,null,"must be in the form of a text string, contained in quotes."]
[null,21,"2008-10-12",1]
[null,22,"2008-08-15",1,null,"If the criterion is simple text, COUNTIF counts the number of matching cells in"]
[null,23,"2008-09-20",1,null,"the range.",null,{"$f":"COUNTIF(N10:N15,\"*mean\")","v":1},{"$f":"showformula(H26)","v":"=COUNTIF(N10:N15,\"*mean\")"}]
[null,24,"2008-12-06",1]
[null,25,"2008-04-03",1,null,"The character \"*\" can be used as a wildcard."]
[null,26,"2008-04-20",1,null,null,null,{"$f":"COUNTIF(N10:N15,\"m*m\")","v":2},{"$f":"showformula(H29)","v":"=COUNTIF(N10:N15,\"m*m\")"},null,null,"\"minimum\" and \"maximum\" both match"]
[null,27,"2008-03-01",1]
[null,28,"2008-04-18",1,null,"If the criterion is a numeric condition, COUNTIF counts the cells satisfying the condition."]
[null,29,"2008-12-25",1,null,null,null,{"$f":"COUNTIF(F14:M14,\"<=10\")","v":6},{"$f":"showformula(H32)","v":"=COUNTIF(F14:M14,\"<=10\")"}]
[null,30,"2008-04-30",1]
[null,31,"2008-07-17",1]
[null,32,"2008-05-17",1,null,null,null,"order","order"]
[null,33,{"$f":"DATE(2008,1,1)+RANDBETWEEN(0,365)","v":"2008-04-23"},{"$f":"COUNTIF($C$4:$C$63,C36)","v":1},null,null,null,"date","cost"]
[null,34,"2008-05-09",1,null,null,"Fred","2008-01-02",120,{"$f":"IF(AND(H37>=$K$44,H37<=$K$45),1,0)","v":1},{"$f":"showformula(J37)","v":"=IF(AND(H37>=$K$44,H37<=$K$45),1,0)"}]
[null,35,"2008-12-07",1,null,null,"Mary","2008-03-12",150,1,"=IF(AND(H38>=$K$44,H38<=$K$45),1,0)"]
[null,36,"2008-05-03",1,null,null,"Bob","2007-09-01",70,0,"=IF(AND(H39>=$K$44,H39<=$K$45),1,0)"]
[null,37,"2008-02-20",1,null,null,"Ellen","2009-01-03",200,0,"=IF(AND(H40>=$K$44,H40<=$K$45),1,0)"]
[null,38,"2008-07-29",1,null,null,"Fred","2007-03-20",180,0,"=IF(AND(H41>=$K$44,H41<=$K$45),1,0)"]
[null,39,"2008-06-08",1,null,null,"George","2008-04-12",100,1,"=IF(AND(H42>=$K$44,H42<=$K$45),1,0)"]
[null,40,"2008-02-04",1]
[null,41,"2008-05-06",1,null,null,null,null,null,"date",{"$f":"DATE(2008,1,1)","v":"2008-01-01"},{"$f":"showformula(K44)","v":"=DATE(2008,1,1)"}]
[null,42,"2008-08-16",1,null,null,null,null,null,"range",{"$f":"DATE(2008,12,31)","v":"2008-12-31"},{"$f":"showformula(K45)","v":"=DATE(2008,12,31)"}]
[null,43,"2008-05-20",1]
[null,44,"2008-12-24",1,null,"There's a similar SUMIF function: =SUMIF(range, criterion, sum-range)"]
[null,45,"2008-03-20",1]
[null,46,"2008-12-29",1,null,null,{"$f":"SUMIF(G37:G42,\"Fred\",I37:I42)","v":300},{"$f":"showformula(G49)","v":"=SUMIF(G37:G42,\"Fred\",I37:I42)"},null,null,"orders placed by Fred"]
[null,47,"2008-11-26",1,null,null,{"$f":"SUMIF(I37:I42,\">=150\",I37:I42)","v":530},{"$f":"showformula(G50)","v":"=SUMIF(I37:I42,\">=150\",I37:I42)"},null,null,"orders of at least $150"]
[null,48,"2008-10-09",1]
[null,49,"2008-12-16",1,null,"Both COUNTIF and SUMIF are limited to testing for a single criterion. If"]
[null,50,"2008-03-11",1,null,"you need to test for multiple criteria (for example, orders placed in 2008),"]
[null,51,"2008-01-17",1,null,"you have two approaches:"]
[null,52,"2008-12-19",1]
[null,53,"2008-05-11",1,null,"Add a new logical column to your data:"]
[null,54,"2008-10-01",1]
[null,55,"2008-10-29",1,null,null,{"$f":"SUM(J37:J42)","v":3},{"$f":"showformula(G58)","v":"=SUM(J37:J42)"},null,null,"number of orders in 2008"]
[null,56,"2008-02-11",1,null,null,{"$f":"SUMPRODUCT(J37:J42,I37:I42)","v":370},{"$f":"showformula(G59)","v":"=SUMPRODUCT(J37:J42,I37:I42)"},null,null,"total cost of orders in 2008"]
[null,57,"2008-12-30",1]
[null,58,"2008-10-15",1,null,"Use array formulae:"]
[null,59,"2008-05-29",1]
[null,60,"2008-02-19",1,null,null,{"$f":"SUM((H37:H42>=K44)*(H37:H42<=K45))","v":3},{"$f":"showformula(G63)","v":"=SUM((H37:H42>=K44)*(H37:H42<=K45))"},null,null,null,"both entered by"]
[null,null,null,null,null,null,{"$f":"SUM((H37:H42>=K44)*(H37:H42<=K45)*I37:I42)","v":370},{"$f":"showformula(G64)","v":"=SUM((H37:H42>=K44)*(H37:H42<=K45)*I37:I42)"},null,null,null,"typing Ctrl+Shift+Enter"]

{"sheet":"Red bag, blue bag","index":2,"n_rows":62,"n_cols":5,"rows_kept":62,"cut":false,"hidden":false,"merged":[[0,1,0,4]]}
[null,"Updating Beliefs"]
[]
[null,"I walk into the room carrying two small bags.  I open both bags, and let you examine the"]
[null,"contents.  You see that one of the bags (call it \"Bag R\") contains 70 red marbles and 30"]
[null,"blue marbles.  The other (Bag B) contains 70 blue marbles and 30 red."]
[]
[null,"I reseal the bags, and quickly slide them back and forth past each other on the table. "]
[null,"(The old \"shell\" game - you completely lose track of which bag is which.)  I then throw one"]
[null,"of the bags out the door, leaving the other in front of you on the table."]
[]
[null,"At this point, you figure it's 50:50 whether Bag R or Bag B is sitting on the table."]
[]
[null,"I loosen the tie at the top of the bag, reach in, pull out a dozen marbles (chosen from the"]
[null,"bag at random), and reseal the bag again.  We look at the marbles, and see that eight are"]
[null,"red, and four are blue."]
[]
[null,"What are the odds that this bag is bag R?"]
[]
[null,"For instance -"]
[]
[null,"even odds:","Betting $1 that it's Bag R against $1 that it's Bag B, you'd be"]
[null,null,"equally happy to be on either side of the bet."]
[null,"2-to-1:","Betting $2 that it's Bag R against $1 that it's Bag B, you'd be"]
[null,null,"equally happy to be on either side of the bet."]
[null,"3-to-1:","Betting $3 that it's Bag R against $1 that it's Bag B, you'd be"]
[null,null,"equally happy to be on either side of the bet."]
[null,"5-to-1:","Betting $5 that it's Bag R against $1 that it's Bag B, you'd be"]
[null,null,"equally happy to be on either side of the bet."]
[null,"10-to-1:","Betting $10 that it's Bag R against $1 that it's Bag B, you'd be"]
[null,null,"equally happy to be on either side of the bet."]
[null,"25-to-1:","Betting $20 that it's Bag R against $1 that it's Bag B, you'd be"]
[null,null,"equally happy to be on either side of the bet."]
[null,"50-to-1:","Betting $50 that it's Bag R against $1 that it's Bag B, you'd be"]
[null,null,"equally happy to be on either side of the bet."]
[]
[]
[null,70,"red balls in bag R"]
[null,30,"blue balls in bag R"]
[]
[null,30,"red balls in bag B"]
[null,70,"blue balls in bag B"]
[]
[null,0.5,"initial probability of bag R being selected"]
[null,{"$f":"1-B43","v":0.5},"initial probability of bag B being selected"]
[]
[null,12,"sample size"]
[]
[]
[null,"red balls\nin sample","Pr(sample|bag R)","Pr(sample|bag B)","Pr(bag R|sample)"]
[null,0,{"$f":"(COMBIN(redsR,B50)*COMBIN(bluesR,sample-B50)/COMBIN(redsR+bluesR,sample))","v":8.2341480979339378E-8},{"$f":"(COMBIN(redsB,B50)*COMBIN(bluesB,sample-B50)/COMBIN(redsB+bluesB,sample))","v":1.0128218629387807E-2},{"$f":"(C50*priorR)/((C50*priorR)+(D50*priorB))","v":8.1298414429203401E-6}]
[null,1,3.6403602117181599E-6,6.1799300111518823E-2,5.8902702426971209E-5]
[null,2,6.907583501735211E-5,0.1642831394631209,4.2029147518363012E-4]
[null,3,7.4558044145713411E-4,0.25136218059931059,2.9573878978543947E-3]
[null,4,5.1089205249846793E-3,0.24629439470013093,2.0321611592154093E-2]
[null,5,2.3456609019060094E-2,0.16263248919881665,0.12605042016806722]
[null,6,7.4116368775502359E-2,7.4116368775502359E-2,0.5]
[null,7,0.16263248919881665,2.3456609019060094E-2,0.87394957983193278]
[null,8,0.24629439470013093,5.1089205249846793E-3,0.97967838840784593]
[null,9,0.25136218059931059,7.4558044145713411E-4,0.99704261210214562]
[null,10,0.1642831394631209,6.907583501735211E-5,0.99957970852481626]
[null,11,6.1799300111518823E-2,3.6403602117181599E-6,0.99994109729757297]
[null,12,1.0128218629387807E-2,8.2341480979339378E-8,0.99999187015855706]

{"sheet":"Let's make a deal!","index":3,"n_rows":46,"n_cols":6,"rows_kept":46,"cut":false,"hidden":false,"merged":[[0,1,0,8]]}
[null,"Let's Make a Deal"]
[]
[null,"Perhaps you remember an old TV game show called \"Let's Make a Deal,\" hosted by"]
[null,"Monty Hall.  Each show ended the same way:  One member of the studio audience was"]
[null,"given a shot at a \"fabulous prize\" (a new car, exotic cruise, or the like)."]
[]
[null,"On the stage were three closed curtains.  One concealed the prize; the other two"]
[null,"concealed garbage (joke prizes, such as \"a pig in a poke,\" not worth very much money)."]
[]
[null,"The contestant selected a curtain, and could have whatever it concealed.  But, before"]
[null,"the curtain was opened, Monty Hall would step in and complicate matters:"]
[]
[null,"No matter which curtain was selected (the prize or garbage), there was always at least"]
[null,"one of the other two which concealed garbage.  The stage crew would signal to Monty,"]
[null,"and he'd open an unselected garbage-hiding curtain."]
[]
[null,"Everyone would see the joke prize, and laugh.  Then, the moment of truth arrived. "]
[null,"Monty Hall would turn to the contestant, and point at the remaining two closed curtains."]
[]
[null,"\"One of these hides the prize; the other hides garbage.  You've already selected this"]
[null,"one.  (He points.)  \"You can have what it conceals, or, if you're having second thoughts,"]
[null,"you can switch your choice, and have what the other conceals instead.\""]
[]
[null,"What would you do?  Stay with your original choice, or switch?  Or does it not really"]
[null,"matter?"]
[]
[]
[null,"Say you start by choosing door A."]
[null,"There are three possibilities: A hides the prize, or B does, or C does."]
[null,"Each possibility has probability 1/3."]
[]
[null,"Now, Monty has two choices in the first case: open B, or open C."]
[null,"In the other cases, he has only one choice: open the unchosen, nonprize curtain."]
[]
[null,"So the progression of events, with associated probabilities, is:"]
[]
[null,null,"prize","opened","probability"]
[null,null,"A","B",{"$f":"1/6","v":0.16666666666666666},"x"]
[null,null,"A","C",{"$f":"1/6","v":0.16666666666666666}]
[null,null,"B","C",{"$f":"1/3","v":0.33333333333333331}]
[null,null,"C","B",{"$f":"1/3","v":0.33333333333333331},"x"]
[]
[null,"Given, for example, that he opens curtain B, then the probability that the"]
[null,"prize is behind curtain A is only (1/6)/(1/6+1/3) = 1/3 = 33.33%"]
[]
[null,"Your chances are twice as good from switching as they are from sticking!"]