document.write( "Question 1205063: You were tasked to program an algorithm for a number generator. The generator will be used by a soda company for their bottlecaps. The soda company will be holding a bottlecap lottery where 5 select bottlecaps will be winning combinations. There would be three characters printed on the bottlecaps with each character having the chance to be repeated. If a second person was to join, how many bottles of Pepsi will the 2nd person need to buy to have a 1% chance at winning. \n" ); document.write( "

Algebra.Com's Answer #848192 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "You were tasked to program an algorithm for a number generator. The generator will be used
\n" ); document.write( "by a soda company for their bottlecaps. The soda company will be holding a bottlecap lottery
\n" ); document.write( "where 5 select bottlecaps will be winning combinations. There would be three characters printed
\n" ); document.write( "on the bottlecaps with each character having the chance to be repeated. If a second person
\n" ); document.write( "was to join, how many bottles of Pepsi will the 2nd person need to buy to have a 1% chance at winning.
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\n" ); document.write( "\n" ); document.write( "To me, the way how this problem is worded in the post, is soup of words without clear meaning
\n" ); document.write( "(total gibberish, in plain English), as well as the solution, given to it in the post by other tutor.\r
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\n" ); document.write( "\n" ); document.write( "Regarding the problem's formulation, I do not understand what the first person is doing
\n" ); document.write( "in this problem. I also do not understand what the generator of numbers is doing in this problem
\n" ); document.write( "and for what reason they tell me about it. From the problem, I do not understand what characters
\n" ); document.write( "do they use.\r
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\n" ); document.write( "\n" ); document.write( "Regarding the solution by the other tutor, I do not understand which characters does he consider (uses),
\n" ); document.write( "how the other tutor interprets the second person's actions, why he consider how many losing are made.\r
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\n" ); document.write( "\n" ); document.write( "So, as given, the problem is a fog and can not be considered as a Math problem.\r
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\n" ); document.write( "\n" ); document.write( "Nevertheless, if to re-formulate this problem totally/radically/cardinally, it is possible to get really
\n" ); document.write( "interesting mathematical problem. I will make this editing right here step by step, then will solve
\n" ); document.write( "the updated/re-modeled problem.\r
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\n" ); document.write( "\n" ); document.write( "First, I will take off a person named \"You\" with his algorithm as an unnecessary item (first sentence).
\n" ); document.write( "Second, I will remove every mention about this generator, together with the second sentence.\r
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\n" ); document.write( "\n" ); document.write( "Now the problem's formulation will be as follows\r
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document.write( "    The soda company will be holding a bottlecap lottery. In each bottlecap, \r\n" );
document.write( "    they will print 3 digits from 0 to 9 inclusive, allowing each digit \r\n" );
document.write( "    to be repeated.  Some combination of 5 selected bottlecaps will be winning. \r\n" );
document.write( "    If a person wants to participate in this lottery, how many bottles of Pepsi \r\n" );
document.write( "    needs he to buy to have a 1% chance of winning.\r\n" );
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\r
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\n" ); document.write( "\n" ); document.write( "Now it is 100% perfect, clean and clear Math problem, and my solution is below.\r
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document.write( "In this problem, the number of all possible different 3-digit numbers on bottlecaps is 10*10*10 = 1000.\r\n" );
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document.write( "It means that 1000 bottles/bottlecaps are in the game. \r\n" );
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document.write( "The number of all possible combinations of 1000 bottlecaps taken 5 at a time is  \r\n" );
document.write( "(the order does not matter).  It is the gross pool of all possible combinations.\r\n" );
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document.write( "One of these combinations is winning.\r\n" );
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document.write( "Let n be the minimal number of bottles for the person to buy in order for to have a pool of combinations  \r\n" );
document.write( ", which is 0.01 of the gross pool  .\r\n" );
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document.write( "In other words, we want to solve this equation for n\r\n" );
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document.write( "     = 0.01.    (1)\r\n" );
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document.write( "It is difficult to solve this equation directly (if possible, at all).  \r\n" );
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document.write( "But it is possible to ESTIMATE \"n\", using well known asymptotic formulas for  .\r\n" );
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document.write( "For great values of \"m\"  and  limited values of \"k\", the following asymptotic formula is valid\r\n" );
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document.write( "         = .      (*)\r\n" );
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document.write( "If to use this formula in equation (1),  it gives\r\n" );
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document.write( "     = 0.01.\r\n" );
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document.write( "Simplify it\r\n" );
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document.write( "     = ,\r\n" );
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document.write( "     = ,\r\n" );
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document.write( "    n = 10^(13/5) = 398.1072.\r\n" );
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document.write( "So, this estimation tells that the number of bottles to buy for the person wishing\r\n" );
document.write( "1% chance to win is 398, or about 400.\r\n" );
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document.write( "At this point the problem is solved completely to that extent \r\n" );
document.write( "as the elementary probability methods allow to do it.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "Thus, what I did in this my post and what is my contribution, is listed here:\r
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\n" ); document.write( "\n" ); document.write( "        - I explained that the original formulation is total gibberish;\r
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\n" ); document.write( "\n" ); document.write( "        - I reformulated the problem and transformed it into the first class probability problem,\r
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\n" ); document.write( "\n" ); document.write( "        - and solved it to the end, providing first class solution.\r
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\n" ); document.write( "\n" ); document.write( "The reference to the formula (*) is this AI Overview\r
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\n" ); document.write( "\n" ); document.write( "https://www.google.com/search?q=asymtotic+formula+for+C%5Bn%5D%5Ek+for+great+n&sca_esv=d2c0a33562fe64ea&rlz=1C1CHBF_enUS1071US1071&sxsrf=ADLYWIJjvf1rB_BUAXepFBBC4xLGziiRpg%3A1735105026743&ei=ApprZ_-FLb2d0PEPytqJaQ&ved=0ahUKEwj_jpH2mcKKAxW9DjQIHUptIg0Q4dUDCBI&uact=5&oq=asymtotic+formula+for+C%5Bn%5D%5Ek+for+great+n&gs_lp=Egxnd3Mtd2l6LXNlcnAiKGFzeW10b3RpYyBmb3JtdWxhIGZvciBDW25dXmsgZm9yIGdyZWF0IG4yCBAAGIAEGKIEMggQABiABBiiBDIIEAAYgAQYogQyCBAAGIAEGKIESJ0gUK8WWK8WcAF4AZABAJgBZ6ABZ6oBAzAuMbgBA8gBAPgBAZgCAqACccICChAAGLADGNYEGEeYAwDiAwUSATEgQIgGAZAGCJIHAzEuMaAH1QQ&sclient=gws-wiz-serp\r
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