document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177711>Question 1177711</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 #850401 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=850401\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step-by-step.\r<BR>\n" );
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document.write( "**1. Define Random Variables**\r<BR>\n" );
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document.write( "* **X:** The outcome of the first roll (X can be 1, 2, 3, 4, 5, or 6).<BR>\n" );
document.write( "* **Y:** The number of wins (getting a \"3\") in the subsequent rolls.\r<BR>\n" );
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document.write( "**2. Probability of Winning on a Single Roll**\r<BR>\n" );
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document.write( "* The probability of rolling a \"3\" on any single roll is 1/6.\r<BR>\n" );
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document.write( "**3. Conditional Expectation**\r<BR>\n" );
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document.write( "We need to find the expected number of wins given the outcome of the first roll.\r<BR>\n" );
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document.write( "* If X = 1, we roll the die once more. The expected number of wins is 1 * (1/6) = 1/6.<BR>\n" );
document.write( "* If X = 2, we roll the die twice more. The expected number of wins is 2 * (1/6) = 2/6.<BR>\n" );
document.write( "* If X = 3, we roll the die three more times. The expected number of wins is 3 * (1/6) = 3/6.<BR>\n" );
document.write( "* If X = 4, we roll the die four more times. The expected number of wins is 4 * (1/6) = 4/6.<BR>\n" );
document.write( "* If X = 5, we roll the die five more times. The expected number of wins is 5 * (1/6) = 5/6.<BR>\n" );
document.write( "* If X = 6, we roll the die six more times. The expected number of wins is 6 * (1/6) = 6/6 = 1.\r<BR>\n" );
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document.write( "**4. Expected Value of X**\r<BR>\n" );
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document.write( "* Since the die is fair, each outcome of the first roll has a probability of 1/6.\r<BR>\n" );
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document.write( "**5. Total Expected Wins**\r<BR>\n" );
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document.write( "We need to find the expected value of Y, which is the sum of the conditional expectations multiplied by the probability of each outcome of X:\r<BR>\n" );
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document.write( "* E(Y) = (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(Y) = (1/36) * (1 + 2 + 3 + 4 + 5 + 6)<BR>\n" );
document.write( "* E(Y) = (1/36) * 21<BR>\n" );
document.write( "* E(Y) = 21/36<BR>\n" );
document.write( "* E(Y) = 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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