document.write( "Question 1196181: There are 15 green apples and 72 red apples in a basket. We keep extracting the apples randomly, one at a time, until we have extracted all 72red apples. What is the probability that the basket is empty when we finish?
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Algebra.Com's Answer #828916 by ikleyn(53354) $\"\"$ $\"About$

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\n" ); document.write( "There are 15 green apples and 72 red apples in a basket.
\n" ); document.write( "We keep extracting the apples randomly, one at a time,
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\n" ); document.write( "What is the probability that the basket is empty when we finish?
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\n" ); document.write( "\n" ); document.write( "            The solution by @greenestamps does not correspond to the posed question.\r
\n" ); document.write( "\n" ); document.write( "            It does not correspond, becouse he considers the strings of the length 72+15 = 87, ONLY.\r
\n" ); document.write( "\n" ); document.write( "            Meanwhile, the process can stop at any number of apples extracted between 72 and 87;\r
\n" ); document.write( "\n" ); document.write( "            the only two conditions are that all 72 red apples are extracted and the last extracted apple is red.\r
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\n" ); document.write( "\n" ); document.write( "            The green apples can still remain in the basket when the process stops.\r
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\n" ); document.write( "\n" ); document.write( "            So, I came to bring another solution.\r
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document.write( "So, all possible outcomes are the strings of the length between 72 and 87 inclusive,\r\n" );
document.write( "that contain 72 \"R\" and such that \"R\" is in the last position.\r\n" );
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document.write( "Let's consider only distinguishable arrangements.\r\n" );
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document.write( "The number of such strings of the length 72 is 1.\r\n" );
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document.write( "The number of such strings of the length 73 is 72  (a unique green apple is in any position from 1 to 72).\r\n" );
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document.write( "The number of such strings of the length 74 is   (two green apples are in any two positions from 1 to 73).\r\n" );
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document.write( "The number of such strings of the length 75 is   (three green apples are in any three positions from 1 to 74).\r\n" );
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document.write( "By continuing this logical reasoning, the total number of all possible outcomes is the sum\r\n" );
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document.write( "    Total = 1 +  +  +  + . . . +  + .\r\n" );
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document.write( "The favorable outcomes are the strings of the length 87 (that contain all the apples) with the red apple in the last position.\r\n" );
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document.write( "The number of such strings is  Favorable = .\r\n" );
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document.write( "Now we should calculate these numbers.\r\n" );
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document.write( "To make it, I used Excel spreadsheet. It gave me Total = 26322377293583700;  Favorable = 21784036380896900.\r\n" );
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document.write( "Now the ANSWER to the problem's question is this ratio  \r\n" );
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document.write( "    P =  =  = 0.8276  (rounded).\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "Again, @greenestamps answered other question, different from the posed in the problem, namely\r
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document.write( "    \"What is the probability that last extracted apple is red given that the basket is empty?\"\r\n" );
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\n" ); document.write( "\n" ); document.write( "although the real question in the problem is \r
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document.write( "    \"What is the probability that the basket is empty when we extract the 72-th red apple?\"\r\n" );
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\n" ); document.write( "\n" ); document.write( "The questions seems to be similar, but in reality they are different, and the logic of solutions is different.\r
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\n" ); document.write( "\n" ); document.write( "At the finish, I have even more interesting message to you.\r
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\n" ); document.write( "\n" ); document.write( "When I calculated the probability according to the @greenestamps solution,
\n" ); document.write( "I got practically the same decimal rounded number of 0.8276.\r
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\n" ); document.write( "\n" ); document.write( "The difference is in further decimal places.\r
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\n" ); document.write( "\n" ); document.write( "So, numerically our answers are very close, but logically the solutions are different.\r
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