{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2025-12-14T01:33:56.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2025-12-14T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":1301,"title":"RISK Calculator - Large Armies, High Accuracy, Fast","description":"This Challenge is to quickly provide the high precision probability of legal RISK battles up to 100 vs 100.  [ Attack \u003e= 2 and Defense \u003e=1 ].\r\n\r\nRelated to  \u003chttp://www.mathworks.com/matlabcentral/cody/problems/1260-risk-board-game-battle-simulation/solutions/map Cody 1260 RISK Board Game Battle Simulation\u003e\r\n\r\n\u003chttp://en.wikipedia.org/wiki/Risk_%28game%29#Official Link to official Risk Rules\u003e\r\n\r\n*Simplified explanation of the dice play:*\r\n  \r\n  Attacker with 2 armies will throw one die.\r\n  Attacker with 3 armies will throw two die.\r\n  Attacker with 4 or more armies will throw three die.\r\n  \r\n  Defense with 1 army will use one die.\r\n  Defense with 2 or more armies will throw 2 die.\r\n  \r\n  The attacker High is compared to the Defender High.\r\n  If Attacker High \u003e Defender High then defender loses 1 army otherwise Attacker loses 1 army. Tie goes to defender.\r\n  If the Defender threw two die and the Attacker threw 2 or more die then the Second Highest of each is compared. \r\nIf Attack \u003e Defense then Defense loses an army otherwise Attack loses an army.\r\nAttack continues until No defenders remain (Win) or Attack is reduced to 1 army (Lose). \r\n\r\n*Input:* a,d where a is number of attacking armies and d is number defending\r\n\r\n*Output:* pwin, the probability of the Attacker Winning\r\n\r\n*Accuracy:* Accurate to +/- 1e-6\r\n\r\n*Scoring:* Time (msec) to solve 10 Battle Scenarios \r\n\r\n\r\n\u003chttp://recreationalmath.com/Risk/  Risk Calculator\u003e","description_html":"\u003cp\u003eThis Challenge is to quickly provide the high precision probability of legal RISK battles up to 100 vs 100.  [ Attack \u003e= 2 and Defense \u003e=1 ].\u003c/p\u003e\u003cp\u003eRelated to  \u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/1260-risk-board-game-battle-simulation/solutions/map\"\u003eCody 1260 RISK Board Game Battle Simulation\u003c/a\u003e\u003c/p\u003e\u003cp\u003e\u003ca href = \"http://en.wikipedia.org/wiki/Risk_%28game%29#Official\"\u003eLink to official Risk Rules\u003c/a\u003e\u003c/p\u003e\u003cp\u003e\u003cb\u003eSimplified explanation of the dice play:\u003c/b\u003e\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003eAttacker with 2 armies will throw one die.\r\nAttacker with 3 armies will throw two die.\r\nAttacker with 4 or more armies will throw three die.\r\n\u003c/pre\u003e\u003cpre class=\"language-matlab\"\u003eDefense with 1 army will use one die.\r\nDefense with 2 or more armies will throw 2 die.\r\n\u003c/pre\u003e\u003cpre class=\"language-matlab\"\u003eThe attacker High is compared to the Defender High.\r\nIf Attacker High \u003e Defender High then defender loses 1 army otherwise Attacker loses 1 army. Tie goes to defender.\r\nIf the Defender threw two die and the Attacker threw 2 or more die then the Second Highest of each is compared. \r\nIf Attack \u003e Defense then Defense loses an army otherwise Attack loses an army.\r\nAttack continues until No defenders remain (Win) or Attack is reduced to 1 army (Lose). \r\n\u003c/pre\u003e\u003cp\u003e\u003cb\u003eInput:\u003c/b\u003e a,d where a is number of attacking armies and d is number defending\u003c/p\u003e\u003cp\u003e\u003cb\u003eOutput:\u003c/b\u003e pwin, the probability of the Attacker Winning\u003c/p\u003e\u003cp\u003e\u003cb\u003eAccuracy:\u003c/b\u003e Accurate to +/- 1e-6\u003c/p\u003e\u003cp\u003e\u003cb\u003eScoring:\u003c/b\u003e Time (msec) to solve 10 Battle Scenarios\u003c/p\u003e\u003cp\u003e\u003ca href = \"http://recreationalmath.com/Risk/\"\u003eRisk Calculator\u003c/a\u003e\u003c/p\u003e","function_template":"function pwin = risk_prob(a, d)\r\n pwin=0;\r\nend","test_suite":"%%\r\nfeval(@assignin,'caller','score',5000); % msec\r\n%%\r\na=[100 99 100 10 9 2 2 10 30 70];\r\nd=[100 100 99 9 10 1 2 2 30 80];\r\ny_c=[0.8079031789315619 0.7888693135658454 0.8230449788340404 0.5580697529719042 0.3798720048109818 0.4166666666666667 0.10609567901234569 0.9901146432872121 0.633266311153744 0.5011352886279803];\r\n\r\ntsum=0;\r\nfor i=1:length(a)\r\n ta=clock;\r\n y=risk_prob(a(i), d(i));\r\n t1=etime(clock,ta)*1000; % time in msec\r\n tsum=tsum+t1;\r\n assert(abs(y - y_c(i)) \u003c= 1e-6,sprintf('A=%i D=%i Expect=%.9f pwin=%.9f',a(i),d(i),y_c(i),y))\r\n fprintf('A %3i  D %3i  Time(msec) %7.3f\\n',a(i),d(i),t1);\r\nend\r\n\r\nfeval(  @assignin,'caller','score',floor(min( 5000,tsum ))  );","published":true,"deleted":false,"likes_count":5,"comments_count":0,"created_by":3097,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":27,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2013-02-25T00:07:52.000Z","updated_at":"2026-02-15T07:40:59.000Z","published_at":"2013-02-25T04:24:12.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis Challenge is to quickly provide the high precision probability of legal RISK battles up to 100 vs 100. [ Attack \u0026gt;= 2 and Defense \u0026gt;=1 ].\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eRelated to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/1260-risk-board-game-battle-simulation/solutions/map\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody 1260 RISK Board Game Battle Simulation\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Risk_%28game%29#Official\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eLink to official Risk Rules\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSimplified explanation of the dice play:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[Attacker with 2 armies will throw one die.\\nAttacker with 3 armies will throw two die.\\nAttacker with 4 or more armies will throw three die.\\n\\nDefense with 1 army will use one die.\\nDefense with 2 or more armies will throw 2 die.\\n\\nThe attacker High is compared to the Defender High.\\nIf Attacker High \u003e Defender High then defender loses 1 army otherwise Attacker loses 1 army. Tie goes to defender.\\nIf the Defender threw two die and the Attacker threw 2 or more die then the Second Highest of each is compared. \\nIf Attack \u003e Defense then Defense loses an army otherwise Attack loses an army.\\nAttack continues until No defenders remain (Win) or Attack is reduced to 1 army (Lose).]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eInput:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e a,d where a is number of attacking armies and d is number defending\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eOutput:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e pwin, the probability of the Attacker Winning\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eAccuracy:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e Accurate to +/- 1e-6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eScoring:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e Time (msec) to solve 10 Battle Scenarios\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"http://recreationalmath.com/Risk/\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eRisk Calculator\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":1301,"title":"RISK Calculator - Large Armies, High Accuracy, Fast","description":"This Challenge is to quickly provide the high precision probability of legal RISK battles up to 100 vs 100.  [ Attack \u003e= 2 and Defense \u003e=1 ].\r\n\r\nRelated to  \u003chttp://www.mathworks.com/matlabcentral/cody/problems/1260-risk-board-game-battle-simulation/solutions/map Cody 1260 RISK Board Game Battle Simulation\u003e\r\n\r\n\u003chttp://en.wikipedia.org/wiki/Risk_%28game%29#Official Link to official Risk Rules\u003e\r\n\r\n*Simplified explanation of the dice play:*\r\n  \r\n  Attacker with 2 armies will throw one die.\r\n  Attacker with 3 armies will throw two die.\r\n  Attacker with 4 or more armies will throw three die.\r\n  \r\n  Defense with 1 army will use one die.\r\n  Defense with 2 or more armies will throw 2 die.\r\n  \r\n  The attacker High is compared to the Defender High.\r\n  If Attacker High \u003e Defender High then defender loses 1 army otherwise Attacker loses 1 army. Tie goes to defender.\r\n  If the Defender threw two die and the Attacker threw 2 or more die then the Second Highest of each is compared. \r\nIf Attack \u003e Defense then Defense loses an army otherwise Attack loses an army.\r\nAttack continues until No defenders remain (Win) or Attack is reduced to 1 army (Lose). \r\n\r\n*Input:* a,d where a is number of attacking armies and d is number defending\r\n\r\n*Output:* pwin, the probability of the Attacker Winning\r\n\r\n*Accuracy:* Accurate to +/- 1e-6\r\n\r\n*Scoring:* Time (msec) to solve 10 Battle Scenarios \r\n\r\n\r\n\u003chttp://recreationalmath.com/Risk/  Risk Calculator\u003e","description_html":"\u003cp\u003eThis Challenge is to quickly provide the high precision probability of legal RISK battles up to 100 vs 100.  [ Attack \u003e= 2 and Defense \u003e=1 ].\u003c/p\u003e\u003cp\u003eRelated to  \u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/1260-risk-board-game-battle-simulation/solutions/map\"\u003eCody 1260 RISK Board Game Battle Simulation\u003c/a\u003e\u003c/p\u003e\u003cp\u003e\u003ca href = \"http://en.wikipedia.org/wiki/Risk_%28game%29#Official\"\u003eLink to official Risk Rules\u003c/a\u003e\u003c/p\u003e\u003cp\u003e\u003cb\u003eSimplified explanation of the dice play:\u003c/b\u003e\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003eAttacker with 2 armies will throw one die.\r\nAttacker with 3 armies will throw two die.\r\nAttacker with 4 or more armies will throw three die.\r\n\u003c/pre\u003e\u003cpre class=\"language-matlab\"\u003eDefense with 1 army will use one die.\r\nDefense with 2 or more armies will throw 2 die.\r\n\u003c/pre\u003e\u003cpre class=\"language-matlab\"\u003eThe attacker High is compared to the Defender High.\r\nIf Attacker High \u003e Defender High then defender loses 1 army otherwise Attacker loses 1 army. Tie goes to defender.\r\nIf the Defender threw two die and the Attacker threw 2 or more die then the Second Highest of each is compared. \r\nIf Attack \u003e Defense then Defense loses an army otherwise Attack loses an army.\r\nAttack continues until No defenders remain (Win) or Attack is reduced to 1 army (Lose). \r\n\u003c/pre\u003e\u003cp\u003e\u003cb\u003eInput:\u003c/b\u003e a,d where a is number of attacking armies and d is number defending\u003c/p\u003e\u003cp\u003e\u003cb\u003eOutput:\u003c/b\u003e pwin, the probability of the Attacker Winning\u003c/p\u003e\u003cp\u003e\u003cb\u003eAccuracy:\u003c/b\u003e Accurate to +/- 1e-6\u003c/p\u003e\u003cp\u003e\u003cb\u003eScoring:\u003c/b\u003e Time (msec) to solve 10 Battle Scenarios\u003c/p\u003e\u003cp\u003e\u003ca href = \"http://recreationalmath.com/Risk/\"\u003eRisk Calculator\u003c/a\u003e\u003c/p\u003e","function_template":"function pwin = risk_prob(a, d)\r\n pwin=0;\r\nend","test_suite":"%%\r\nfeval(@assignin,'caller','score',5000); % msec\r\n%%\r\na=[100 99 100 10 9 2 2 10 30 70];\r\nd=[100 100 99 9 10 1 2 2 30 80];\r\ny_c=[0.8079031789315619 0.7888693135658454 0.8230449788340404 0.5580697529719042 0.3798720048109818 0.4166666666666667 0.10609567901234569 0.9901146432872121 0.633266311153744 0.5011352886279803];\r\n\r\ntsum=0;\r\nfor i=1:length(a)\r\n ta=clock;\r\n y=risk_prob(a(i), d(i));\r\n t1=etime(clock,ta)*1000; % time in msec\r\n tsum=tsum+t1;\r\n assert(abs(y - y_c(i)) \u003c= 1e-6,sprintf('A=%i D=%i Expect=%.9f pwin=%.9f',a(i),d(i),y_c(i),y))\r\n fprintf('A %3i  D %3i  Time(msec) %7.3f\\n',a(i),d(i),t1);\r\nend\r\n\r\nfeval(  @assignin,'caller','score',floor(min( 5000,tsum ))  );","published":true,"deleted":false,"likes_count":5,"comments_count":0,"created_by":3097,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":27,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2013-02-25T00:07:52.000Z","updated_at":"2026-02-15T07:40:59.000Z","published_at":"2013-02-25T04:24:12.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis Challenge is to quickly provide the high precision probability of legal RISK battles up to 100 vs 100. [ Attack \u0026gt;= 2 and Defense \u0026gt;=1 ].\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eRelated to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/1260-risk-board-game-battle-simulation/solutions/map\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eCody 1260 RISK Board Game Battle Simulation\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Risk_%28game%29#Official\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eLink to official Risk Rules\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSimplified explanation of the dice play:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[Attacker with 2 armies will throw one die.\\nAttacker with 3 armies will throw two die.\\nAttacker with 4 or more armies will throw three die.\\n\\nDefense with 1 army will use one die.\\nDefense with 2 or more armies will throw 2 die.\\n\\nThe attacker High is compared to the Defender High.\\nIf Attacker High \u003e Defender High then defender loses 1 army otherwise Attacker loses 1 army. Tie goes to defender.\\nIf the Defender threw two die and the Attacker threw 2 or more die then the Second Highest of each is compared. \\nIf Attack \u003e Defense then Defense loses an army otherwise Attack loses an army.\\nAttack continues until No defenders remain (Win) or Attack is reduced to 1 army (Lose).]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eInput:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e a,d where a is number of attacking armies and d is number defending\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eOutput:\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e pwin, the probability of the 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