document.write( "Question 1209959: A club buys tickets for 15 seats in a row for a local production and then sends the tickets randomly to the 15 people who ordered seats. Of these 15 people, 9 are actors and 6 are backstage workers. Find the probability that: a. An actor and backstage worker will receive tickets sitting together. b. The actors will be seated together, and the backstage workers will be seated together. \n" ); document.write( "

Algebra.Com's Answer #851983 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "A club buys tickets for 15 seats in a row for a local production and then sends the tickets randomly
\n" ); document.write( "to the 15 people who ordered seats. Of these 15 people, 9 are actors and 6 are backstage workers.
\n" ); document.write( "Find the probability that:
\n" ); document.write( "(a) An actor and backstage worker will receive tickets sitting together.
\n" ); document.write( "(b) The actors will be seated together, and the backstage workers will be seated together.
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "- - - - - To me, this problem and this case are EXCEPTIONALLY INTERESTING. - - - - - - -
\n" ); document.write( "- - - - - - - - - - - - - - - - - - Below I will explain WHY. - - - - - - - - - - - - - - - - - - - - - - - - \r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Question (a), if to read it literally as it is written in the post, asks to find
\n" ); document.write( "the probability that some of 9 actors sits next to some of 6 backstage workers.\r
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\n" ); document.write( "\n" ); document.write( "            The answer is OBVIOUS:   this probability is 100%.\r
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\n" ); document.write( "\n" ); document.write( "Indeed, having 15 seats in a row, 9 actors and 6 backstage workers, it is IMPOSSIBLE to place
\n" ); document.write( "them in a way that no actor sits next to some backstage worker.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The answer is at the level of common sense and at the level accessible to a kindergarten child.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "@CPhill, which is a synonym to Artificial Intelligence, in his post starts create
\n" ); document.write( "irrelevant reasoning and CAN NOT produce right answer.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From it, I conclude that this AI is not able to make \r
\n" ); document.write( "\n" ); document.write( "OBVIOUS elementary basic logical implications, at all.\r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "         For me, it is 100% proof that this concrete AI, at its stage of development,
\n" ); document.write( "         is not able to think, at all, and does not possess elementary common sense
\n" ); document.write( "                                    even of the level of a kindergarten child.\r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "It is WHY this case and this post by @CPhill are interesting to me.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "It is why this case should be interesting to you, my reader.\r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "- - - - - - - - Now I will return to the problem itself - - - - - - - -\r
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\n" ); document.write( "\n" ); document.write( "- - - - - - - - - - - - - - - - Part (a) - - - - - - - - - - - - - - - - \r
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\n" ); document.write( "\n" ); document.write( "I don't know, WHY question (a) in this post is posed this way.
\n" ); document.write( "It may be a Math joke. It may be a TRAP.
\n" ); document.write( "It may be, that the author/(the creator or the visitor) was intended to ask what is the probability
\n" ); document.write( "that some PARTICULAR actor A will sit next to some PARTICULAR backstage worker B.\r
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\n" ); document.write( "\n" ); document.write( "If so, then the solution is as follows:\r
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\r\n" );
document.write( "    Then we consider this pair (A,B) as one unit, and we have then 14 possible positions for this unit\r\n" );
document.write( "    in the row of 15 seats. We also have 15! permutations for 15 persons, in all,\r\n" );
document.write( "    and 2 possible permutations inside the pair (A,B).\r\n" );
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document.write( "    Therefore, the probability that particular A and particular B sit together is\r\n" );
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document.write( "       P =  =  =  = 2.14121E-11.\r\n" );
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\r
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\n" ); document.write( "\n" ); document.write( "- - - - - - - - - - - - - - - - Part (b) - - - - - - - - - - - - - - - - \r
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\n" ); document.write( "\n" ); document.write( "Question (b) asks to find the probability that the actors seat together, and the backstage workers seat together.\r
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\r\n" );
document.write( "    Then we have a compact group of 9 actors with 9! permutations inside this group,\r\n" );
document.write( "    and a compact group of 6 backstage workers with 6! permutations inside this group.\r\n" );
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document.write( "    The answer to question (b) is\r\n" );
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document.write( "          P =  =  =  = 0.0003996.\r\n" );
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document.write( "The factor  ' 2 '   in the numerator is to account for two different possible placements \r\n" );
document.write( "of the group of actors and the group of backstage workers.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "This problem, in its part (a), was a crucial TEST for the AI,\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " and the AI $\"highlight%28highlight%28failed%29%29\"$ this test.\r
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\n" ); document.write( "\n" ); document.write( "---------------------------------\r
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\n" ); document.write( "\n" ); document.write( "This story has interesting continuation.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "On September 2, 2025, I posted this problem to the Google AI, again.\r
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\n" ); document.write( "\n" ); document.write( "This time, it produced correct, perfect and ideal solution.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Thanks to god, this AI is able to learn from the Internet.\r
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\n" ); document.write( "\n" ); document.write( "It shows how it is important for this AI to have an adequate qualified tutor/teacher/mentor.\r
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