[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-571372-105":53,"doc-detail-571372-en":127},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":120,"head_meta":122,"extra_data":124,"updated_unix":126},105,"en","probabilities-of-poker-hands-with-variations-poker-hands","Probabilities of Poker Hands with Variations - Poker Hands","","Derives the probabilities of different poker hand types in five-card stud, starting from the core hand taxonomy (royal flush through high card) and computing exact counts based on rank and suit combinations. Analyzes how those probabilities change when jokers and wild cards are introduced, and extends the study to Texas Hold em by examining winning probabilities across example games. Includes supporting definitions for poker variants (stud, draw, community cards) and uses combinatorial counting to convert hand counts into probabilities.",{"@graph":63,"@context":119},[64,80,102],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":11,"@type":70,"position":76},"https://docshare.wps.com/template/presentations/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/probabilities-of-poker-hands-with-variations-poker-hands/571372/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/probabilities-of-poker-hands-with-variations-poker-hands/571372.png","ImageObject",442,249,{"name":88,"@type":89},"DoDO","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-20","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":101},"InteractionCounter",{"@type":100},"ViewAction",9,{"@type":103,"mainEntity":104},"FAQPage",[105,111,115],{"name":106,"@type":107,"acceptedAnswer":108},"What are the ten standard poker hand categories used in the analysis?","Question",{"text":109,"@type":110},"The document enumerates ten hands: Royal Flush, Straight Flush, Four-of-a-Kind (4OAK), Full House, Flush, Straight, Three-of-a-Kind (3OAK), Two Pair, One Pair, and High Card.","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"How does the paper define five-card stud and set up the probability calculation?",{"text":114,"@type":110},"In five-card stud, each player is dealt five cards to form the best five-card hand. The paper uses combinatorial counts from the 52-card deck to evaluate how many hands correspond to each category.",{"name":116,"@type":107,"acceptedAnswer":117},"How do jokers and wild cards affect the reported probabilities?",{"text":118,"@type":110},"The document states that it derives probabilities in five-card stud and then analyzes how those probabilities change when jokers and wild cards are included.","https://schema.org",{"og:url":78,"og:type":121,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":123,"canonical":78},"index,follow",{"doc_id":125,"site_id":56},571372,1792033836,{"code":4,"msg":5,"data":128},{"doc_id":125,"user_id":129,"nickname":88,"user_avatar":130,"doc_module":9,"category_id":8,"category_name":11,"doc_title":59,"doc_description":61,"doc_content":131,"file_id":132,"file_url":133,"file_type":134,"file_size":135,"view_count":101,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":136,"language":137,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":138,"faqs":139,"seo_title":140,"seo_description":61,"update_tm":141,"read_time":142},962093270745,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Probabilities of Poker Hands with  \nVariations  \nJeff Duda  \nAcknowledgements:  \nBrian Alspach and Yiu Poon for providing a means to check my numbers  \nPoker is one of the many games involving the use of a 52-card deck of playing cards. The 52 cards are categorized by 13 ranks from Two through Ace (Aces can be counted as both higher than King and lower than Two when needed, but can only count as one at a time in a hand), and by four suits: diamonds, hearts, spades, and clubs. In the game of poker, players attempt to assemble the best five-card hand according to the definitions of each hand that can be made.  \nThere are ten hands that can be made:  \n1) Royal Flush – all five cards are of the same suit and are of the sequence  \n10 – J – Q – K – A  \n2) Straight Flush – all five cards are of the same suit and are sequential in rank (note that a royal flush is simply the highest-ranked straight flush)  \n3) Four-of-a-Kind (which will be abbreviated in this paper as 4OAK)– a hand where four cards are all of the same rank  \n4) Full House – a hand consisting of one pair and a three-of-a-kind of a different rank than the pair  \n5) Flush – all five cards are of the same suit but not all sequential in rank  \n6) Straight – all five cards are sequential in rank but are not all of the same suit  \n7) Three-of-a-Kind (which will be abbreviated as 3OAK)– a hand where three cards are all of the same rank and the other two are each of different ranks from the 3OAK and each other  \n8) Two Pair – two pairs of two cards of the same rank (the ranks of each pair are different in rank, obviously, to avoid a 4OAK)  \n9) One Pair – only two cards of the five are of the same rank with the other three cards all having different ranks from each other and from that of the pair  \n10) High Card – a hand in which no better hand was made (i.e. , one in which each card is of a different rank than any other card and not all five are of the same suit or sequential in rank  \nPoker games have many variations, some of which will be investigated here. One such variation is “stud” poker in which a player must hold all the cards he/she is given. This is opposed to “draw” poker in which a player can draw any number of replacement cards after being dealt an initial five in the attempt to improve his/her hand. Texas Hold em is another variation in which each player is only dealt two cards to themselves, but through the course of the betting rounds a total of five cards are dealt as “community”cards that any player can use with any combination of their two to make the best fivecard hand possible. Other variations include the use of jokers and wild cards. In this paper I will derive the probabilities of being dealt one of the given hands in five-card stud poker and how those probabilities change when jokers and wild cards are included. I will also analyze Texas Hold em and derive the probability of a given hand winning throughout the course of a few example games.  \nFive-Card Stud  \nIn five-card stud each player is dealt five cards to make the best five-card hand possible. Since there are 52 cards in the deck, then there are 52C51 = 2,598,960 possible combinations of five-card hands possible. I will evaluate the numbers of hands in the typical order of rank of each hand, starting with straight flushes (since a royal flush is just the highest-ranked straight flush I will include it in the discussion of straight flushes, but give it no additional importance) .  \nStraight Flush  \nTo have a straight flush the hand must consist of all five cards being of the same suit and all in numerical order. There are 10 possible sequences: A – 5, 2 – 6,…, 9 – K, and 10 – A. Since there are 4 suits, then the number of straight flushes possible is just 10 * 4 = 40, with the highest four (each a straight flush 10 – A of one of the four suits)  \nbeing royal flushes.  \nFour-of-a-Kind (4OAK)  \nTo have a 4OAK the hand must include all of the cards of one of the 13 available ranks plus one additional ","cbCaipZcdhJbiGZc","https://ap.wps.com/l/cbCaipZcdhJbiGZc","pdf",97423,20,"English","# Poker hand definitions\n## Poker variations\n# Five-card stud\n## Straight flush\n## Four-of-a-kind (4OAK)\n## Full house\n## Flush\n## Straight\n## Three-of-a-kind (3OAK)\n## Two pair\n## One pair\n## High card","[{\"question\":\"What are the ten standard poker hand categories used in the analysis?\",\"answer\":\"The document enumerates ten hands: Royal Flush, Straight Flush, Four-of-a-Kind (4OAK), Full House, Flush, Straight, Three-of-a-Kind (3OAK), Two Pair, One Pair, and High Card.\"},{\"question\":\"How does the paper define five-card stud and set up the probability calculation?\",\"answer\":\"In five-card stud, each player is dealt five cards to form the best five-card hand. The paper uses combinatorial counts from the 52-card deck to evaluate how many hands correspond to each category.\"},{\"question\":\"How do jokers and wild cards affect the reported probabilities?\",\"answer\":\"The document states that it derives probabilities in five-card stud and then analyzes how those probabilities change when jokers and wild cards are included.\"}]","Probabilities of Poker Hands with Variations - Poker Hands | PDF",1791914522,7]