{"id":"https://openalex.org/W7214315432","doi":"https://doi.org/10.48550/arxiv.2609.29596","title":"How many cards, until the first ace: variations, extensions, lachrymae, confidence, Dirichlets","display_name":"How many cards, until the first ace: variations, extensions, lachrymae, confidence, Dirichlets","publication_year":2026,"publication_date":"2026-08-27","ids":{"openalex":"https://openalex.org/W7214315432","doi":"https://doi.org/10.48550/arxiv.2609.29596"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.29596","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.29596","pdf_url":"https://arxiv.org/pdf/2609.29596","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.29596","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5024405571","display_name":"Nils Lid Hjort","orcid":"https://orcid.org/0000-0002-0621-032X","observed_orcids":["https://orcid.org/0000-0002-0621-032X"]},"institutions":[{"id":"https://openalex.org/I184942183","display_name":"University of Oslo","ror":"https://ror.org/01xtthb56","country_code":"NO","type":"education","lineage":["https://openalex.org/I184942183"]}],"countries":["NO"],"is_corresponding":true,"raw_author_name":"Hjort, Nils Lid","raw_affiliation_strings":["Department of Mathematics , University of Oslo"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , University of Oslo","institution_ids":["https://openalex.org/I184942183"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5024405571"],"corresponding_institution_ids":["https://openalex.org/I184942183"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T13500","display_name":"Probability and Statistical Research","score":0.9721999764442444,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T13500","display_name":"Probability and Statistical Research","score":0.9721999764442444,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10968","display_name":"Statistical Distribution Estimation and Applications","score":0.023399999365210533,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T13546","display_name":"Census and Population Estimation","score":0.0019000000320374966,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/waiting-time-distribution","display_name":"waiting time distribution","score":0.7770000100135803},{"id":"https://openalex.org/keywords/expected-waiting-time","display_name":"expected waiting time","score":0.6549999713897705},{"id":"https://openalex.org/keywords/uniform-spacings","display_name":"uniform spacings","score":0.46399998664855957},{"id":"https://openalex.org/keywords/dirichlet-distribution","display_name":"Dirichlet distribution","score":0.8119999766349792},{"id":"https://openalex.org/keywords/order-statistics","display_name":"order statistics","score":0.8159999847412109},{"id":"https://openalex.org/keywords/confidence-distributions","display_name":"confidence distributions","score":0.7639999985694885},{"id":"https://openalex.org/keywords/population-size-estimation","display_name":"population size estimation","score":0.7749999761581421},{"id":"https://openalex.org/keywords/coupon-collector-s-problem","display_name":"coupon collector problem","score":0.515999972820282}],"concepts":[{"id":"https://openalex.org/C2908647359","wikidata":"https://www.wikidata.org/wiki/Q2625603","display_name":"Population","level":2,"score":0.49230000376701355},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.4851999878883362},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.48019999265670776},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.4693000018596649},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4404999911785126},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.40059998631477356},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.352400004863739},{"id":"https://openalex.org/C2776214188","wikidata":"https://www.wikidata.org/wiki/Q408386","display_name":"Inference","level":2,"score":0.33799999952316284},{"id":"https://openalex.org/C2776291640","wikidata":"https://www.wikidata.org/wiki/Q2912517","display_name":"Value (mathematics)","level":2,"score":0.33379998803138733},{"id":"https://openalex.org/C149782125","wikidata":"https://www.wikidata.org/wiki/Q160039","display_name":"Econometrics","level":1,"score":0.3124000132083893},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.30970001220703125},{"id":"https://openalex.org/C169214877","wikidata":"https://www.wikidata.org/wiki/Q981016","display_name":"Dirichlet distribution","level":3,"score":0.29919999837875366},{"id":"https://openalex.org/C149441793","wikidata":"https://www.wikidata.org/wiki/Q200726","display_name":"Probability distribution","level":2,"score":0.26840001344680786},{"id":"https://openalex.org/C44082924","wikidata":"https://www.wikidata.org/wiki/Q1767128","display_name":"Order statistic","level":2,"score":0.2678999900817871},{"id":"https://openalex.org/C192065140","wikidata":"https://www.wikidata.org/wiki/Q1147928","display_name":"Multinomial distribution","level":2,"score":0.2678999900817871},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.2542000114917755}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.29596","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.29596","pdf_url":"https://arxiv.org/pdf/2609.29596","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.29596","doi":"https://doi.org/10.48550/arxiv.2609.29596","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.29596","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.29596","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.29596","pdf_url":"https://arxiv.org/pdf/2609.29596","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214315432.pdf","grobid_xml":"https://content.openalex.org/works/W7214315432.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4389674530","https://openalex.org/W7129101393","https://openalex.org/W7213263400","https://openalex.org/W7098997447","https://openalex.org/W2088554512","https://openalex.org/W2034953682","https://openalex.org/W28673774","https://openalex.org/W2027586790","https://openalex.org/W4391887668","https://openalex.org/W2036704322"],"abstract_inverted_index":{"From":[0],"a":[1,93,157,189],"deck":[2,134],"of":[3,38,44,86,129,164,192],"cards,":[4],"how":[5,173],"many":[6],"cards":[7,39],"do":[8],"I":[9,17,58,71],"need":[10,160,184],"to":[11,61,103,161],"draw,":[12],"until":[13,166],"the":[14,19,36,42,49,74,79,81,83,107,126,133,142,169,179],"first":[15,170],"ace?":[16],"identify":[18],"distribution":[20,95],"for":[21,48,78,98,106,114,121,139],"this":[22,55],"waiting":[23],"time":[24],"$T$,":[25],"and":[26,40,65,76,117,137,159,188],"its":[27],"satisfyingly":[28],"nice":[29],"expected":[30],"value":[31],"${\\rm":[32],"E}\\,T=(N+1)/(n+1)$,":[33],"with":[34,88,100],"$N$":[35,146],"number":[37,43,127],"$n$":[41,128,148],"aces;":[45],"hence":[46],"$53/5=10.6$":[47],"standard":[50],"setup.":[51],"After":[52],"having":[53],"solved":[54],"Question":[56],"One":[57],"go":[59],"on":[60],"certain":[62],"alternative":[63],"solutions":[64],"extensions,":[66],"involving":[67],"e.g.":[68],"Beta":[69],"approximations.":[70],"also":[72],"consider":[73],"distributions":[75,120],"means":[77],"2nd,":[80],"3rd,":[82],"4th":[84],"occurrences":[85],"aces,":[87,130],"generalisations,":[89],"where":[90,123],"there":[91,177],"is":[92,112,149],"Dirichlet":[94],"in":[96,156,178],"wait":[97],"us,":[99],"further":[101],"links":[102],"order":[104],"statistics":[105],"uniform.":[108],"Furthermore,":[109],"an":[110,186],"apparatus":[111],"developed":[113],"obtaining":[115],"estimators":[116],"full":[118],"confidence":[119],"applications":[122],"one":[124],"knows":[125],"but":[131],"not":[132],"size":[135,145],"$N$;":[136],"correspondingly":[138],"inference":[140],"about":[141],"unknown":[143],"population":[144],"when":[147],"known.":[150],"If":[151],"you":[152],"have":[153],"1000":[154],"people":[155],"room,":[158],"interview":[162],"11":[163],"them":[165],"you've":[167],"found":[168],"left-handed":[171,175],"person,":[172],"may":[174],"are":[176],"room":[180],"--":[181],"here":[182],"we":[183],"both":[185],"estimate":[187],"clear":[190],"measure":[191],"uncertainty.":[193]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-09-26T00:00:00"}