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 #850361 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850361\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step by step.\r<BR>\n" );
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document.write( "**Understanding the Experiment**\r<BR>\n" );
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document.write( "1.  **First Roll:** Roll a six-sided die once. Let the outcome be X. X can be 1, 2, 3, 4, 5, or 6.<BR>\n" );
document.write( "2.  **Subsequent Rolls:** Roll the die X more times.<BR>\n" );
document.write( "3.  **Wins:** A win occurs if a \"3\" is rolled in any of the subsequent rolls.<BR>\n" );
document.write( "4.  **Goal:** Find the expected number of wins.\r<BR>\n" );
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document.write( "**Calculating Probabilities and Expected Wins**\r<BR>\n" );
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document.write( "1.  **Probability of First Roll Outcomes:**<BR>\n" );
document.write( "    * P(X = 1) = P(X = 2) = P(X = 3) = P(X = 4) = P(X = 5) = P(X = 6) = 1/6\r<BR>\n" );
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document.write( "2.  **Probability of a Win in a Single Roll:**<BR>\n" );
document.write( "    * P(Win) = 1/6<BR>\n" );
document.write( "    * P(No Win) = 5/6\r<BR>\n" );
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document.write( "3.  **Expected Wins Given the First Roll (X):**<BR>\n" );
document.write( "    * Let W be the number of wins.<BR>\n" );
document.write( "    * If X = 1, W ~ Bernoulli(1/6), E[W|X=1] = 1/6<BR>\n" );
document.write( "    * If X = 2, W ~ Binomial(2, 1/6), E[W|X=2] = 2 * (1/6) = 2/6<BR>\n" );
document.write( "    * If X = 3, W ~ Binomial(3, 1/6), E[W|X=3] = 3 * (1/6) = 3/6<BR>\n" );
document.write( "    * If X = 4, W ~ Binomial(4, 1/6), E[W|X=4] = 4 * (1/6) = 4/6<BR>\n" );
document.write( "    * If X = 5, W ~ Binomial(5, 1/6), E[W|X=5] = 5 * (1/6) = 5/6<BR>\n" );
document.write( "    * If X = 6, W ~ Binomial(6, 1/6), E[W|X=6] = 6 * (1/6) = 6/6 = 1\r<BR>\n" );
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document.write( "4.  **Expected Number of Wins (E[W]):**<BR>\n" );
document.write( "    * E[W] = Σ [E[W|X=x] * P(X=x)]<BR>\n" );
document.write( "    * E[W] = (1/6 * 1/6) + (2/6 * 1/6) + (3/6 * 1/6) + (4/6 * 1/6) + (5/6 * 1/6) + (6/6 * 1/6)<BR>\n" );
document.write( "    * E[W] = (1/36) + (2/36) + (3/36) + (4/36) + (5/36) + (6/36)<BR>\n" );
document.write( "    * E[W] = (1 + 2 + 3 + 4 + 5 + 6) / 36<BR>\n" );
document.write( "    * E[W] = 21 / 36<BR>\n" );
document.write( "    * E[W] = 7 / 12\r<BR>\n" );
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document.write( "**Therefore, the expected number of wins from this experiment is 7/12.**<BR>\n" );
document.write( " </SPAN>\n" );
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