document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178071>Question 1178071</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Roll a die once. Then roll it as many times as the outcome from the first roll. Getting the special number \"3\" on any roll means a win. What is the expected number of wins from this experiment?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850379 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52945)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850379\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "Roll a die once. Then roll it as many times as the outcome from the first roll. <BR>\n" );
document.write( "Getting the special number \"3\" on any roll means a win. <BR>\n" );
document.write( "What is the expected number of wins from this experiment? <BR>\n" );
document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Regarding this problem, I have two notices.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "First notice is that the problem's formulation is mathematically incomplete.<BR>\n" );
document.write( "To be complete, it should say \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;\"Getting the special number \"3\" on any roll means a win and stopping further rolling\".\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This correction is almost obvious, but it is important in the analysis.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Second notice is that the solution in the post by @CPhill is INCORRECT.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "It is INCORRECT, since there are errors in his analysis.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Below I will show these errors, but I will not provide a whole solution with complete corrections.\r<BR>\n" );
document.write( "\n" );
document.write( "Why ?  - - - - Because @CPhill is a pseudonym for the artificial intelligence,<BR>\n" );
document.write( "and this solution belongs not to @CPhill, but the artificial intelligence.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So, for AI it will be just a great benefit to get my pointing to their error.<BR>\n" );
document.write( "I myself have no any desire to work here for or instead of AI.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So, I will copy-paste here the part of the @CPhill's text with pointing the errors.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "3.  **Expected Wins Given the First Roll (X):**\r\n" );
document.write( "    * Let W be the number of wins.\r\n" );
document.write( "\r\n" );
document.write( "    * If X = 1, W ~ Bernoulli(1/6), E[W|X=1] = 1/6\r\n" );
document.write( "\r\n" );
document.write( "    * If X = 2, W ~ Binomial(2, 1/6), E[W|X=2] = 2 * (1/6) = 2/6   <<<---=== error. The winning \"3\" can be obtained at                  \r\n" );
document.write( "                                                                                    the first of the two rolls (and then the game stops),              \r\n" );
document.write( "                                                                                    or at the second roll.  It should be correctly counted.\r\n" );
document.write( "\r\n" );
document.write( "    * If X = 3, W ~ Binomial(3, 1/6), E[W|X=3] = 3 * (1/6) = 3/6   <<<---=== error.  The case of getting X = 3 was just analyzed above,\r\n" );
document.write( "                                                                                     and it was just led to a stop/break earlier.\r\n" );
document.write( "\r\n" );
document.write( "    * If X = 4, W ~ Binomial(4, 1/6), E[W|X=4] = 4 * (1/6) = 4/6   <<<---=== similar error: outcome \"3\" can be obtained in any of 4 rolls,\r\n" );
document.write( "                                                                                            leading to stop. It should be accounted.\r\n" );
document.write( "\r\n" );
document.write( "    * If X = 5, W ~ Binomial(5, 1/6), E[W|X=5] = 5 * (1/6) = 5/6   <<<---=== similar error\r\n" );
document.write( "\r\n" );
document.write( "    * If X = 6, W ~ Binomial(6, 1/6), E[W|X=6] = 6 * (1/6) = 6/6 = 1    <<<---=== similar error.\r\n" );
document.write( "</PRE>\r<BR>\n" );
document.write( "\n" );
document.write( "So, this artificial intelligence should fix his brain accordingly and consistently.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "////////////////////////////////////\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Regarding the post by @CPhill . . . \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Keep in mind that @CPhill is a pseudonym for the Google artificial intelligence.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The artificial intelligence is like a baby now. It is in the experimental stage <BR>\n" );
document.write( "of development and can make mistakes and produce nonsense without any embarrassment.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It has no feeling of shame - it is shameless.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This time, again, &nbsp;it made an error.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Although the @CPhill' solution are copy-paste &nbsp;Google &nbsp;AI solutions, &nbsp;there is one essential difference.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Every time, &nbsp;Google &nbsp;AI &nbsp;makes a note at the end of its solutions that &nbsp;Google &nbsp;AI &nbsp;is experimental<BR>\n" );
document.write( "and can make errors/mistakes.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "All @CPhill' solutions are copy-paste of &nbsp;Google &nbsp;AI &nbsp;solutions, with one difference:<BR>\n" );
document.write( "@PChill never makes this notice and never says that his solutions are copy-past that of Google.<BR>\n" );
document.write( "So, he NEVER SAYS TRUTH.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Every time, &nbsp;@CPhill embarrassed to tell the truth.\r<BR>\n" );
document.write( "\n" );
document.write( "But I am not embarrassing to tell the truth, &nbsp;as it is my duty at this forum.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "And the last my comment.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "When you obtain such posts from @CPhill, &nbsp;remember, &nbsp;that &nbsp;NOBODY &nbsp;is responsible for their correctness, <BR>\n" );
document.write( "until the specialists and experts will check and confirm their correctness.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Without it, &nbsp;their reliability is &nbsp;ZERO and their creadability is &nbsp;ZERO, &nbsp;too.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Hello, @CPhill, don't you think, that it is just a time came to express your thanks to me <BR>\n" );
document.write( "for my tireless work on finding, identifying, pointing and correcting your errors ?\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Otherwise, I feel myself uncomfortably about your bad manners.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );