document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192025>Question 1192025</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A received 6 good and 6 bad messages, accidentally deleted one, added one good and sent to B. B also accidentally deleted one message and sent one good message to C. <BR>\n" );
document.write( "a) what is the probability that C will receive less bad messages than A?  <BR>\n" );
document.write( "b) what is the probability that C will receive fewer bad messages than A if B accidentally deleted one and added one bad message? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849090 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</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=849090\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to break down this probability problem:\r<BR>\n" );
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document.write( "**a) B deletes a message (any message) and adds a good one:**\r<BR>\n" );
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document.write( "* **A's messages:** 6 good, 6 bad (12 total)<BR>\n" );
document.write( "* **B's messages:**  B receives 12, deletes 1, and adds 1 good.  There are three scenarios for the deleted message from A:<BR>\n" );
document.write( "    * **Scenario 1 (Deleted Good):** B receives 5 good, 6 bad.  Adds 1 good = 6 good, 6 bad.<BR>\n" );
document.write( "    * **Scenario 2 (Deleted Bad):** B receives 6 good, 5 bad. Adds 1 good = 7 good, 5 bad.<BR>\n" );
document.write( "* **C's messages:** C receives 12, deletes 1, and adds 1 good. Again, two scenarios for the deleted message from B:<BR>\n" );
document.write( "    * **Scenario 1.1 (Deleted Good from B - Scenario 1):** C receives 5 good, 6 bad. Adds 1 good = 6 good, 6 bad.<BR>\n" );
document.write( "    * **Scenario 1.2 (Deleted Bad from B - Scenario 1):** C receives 6 good, 5 bad. Adds 1 good = 7 good, 5 bad.<BR>\n" );
document.write( "    * **Scenario 2.1 (Deleted Good from B - Scenario 2):** C receives 6 good, 5 bad. Adds 1 good = 7 good, 5 bad.<BR>\n" );
document.write( "    * **Scenario 2.2 (Deleted Bad from B - Scenario 2):** C receives 7 good, 4 bad. Adds 1 good = 8 good, 4 bad.\r<BR>\n" );
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document.write( "* **Probability Calculation:** Each of A's deleted message scenarios has a 1/2 probability. From there, each of B's deleted message scenarios has a 1/2 probability. So each final outcome has a 1/4 probability.<BR>\n" );
document.write( "    * C has 6 bad messages (Scenario 1.1): Probability = 1/4<BR>\n" );
document.write( "    * C has 5 bad messages (Scenarios 1.2 & 2.1): Probability = 1/4 + 1/4 = 1/2<BR>\n" );
document.write( "    * C has 4 bad messages (Scenario 2.2): Probability = 1/4\r<BR>\n" );
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document.write( "* **C has *less* bad messages than A (6):** This includes the cases where C has 5 or 4 bad messages.<BR>\n" );
document.write( "* **Probability (C < A) = 1/2 + 1/4 = 3/4 = 0.75**\r<BR>\n" );
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document.write( "**b) B deletes a bad message and adds a bad one:**\r<BR>\n" );
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document.write( "* **A's messages:** 6 good, 6 bad<BR>\n" );
document.write( "* **B's messages:** B receives 12, deletes 1 bad, adds 1 bad. B has 6 good, 6 bad messages.<BR>\n" );
document.write( "* **C's messages:** C receives 12, deletes 1. Two scenarios:<BR>\n" );
document.write( "    * **Scenario 1 (Deleted Good from B):** C receives 5 good, 6 bad. Adds 1 good = 6 good, 6 bad.<BR>\n" );
document.write( "    * **Scenario 2 (Deleted Bad from B):** C receives 6 good, 5 bad. Adds 1 good = 7 good, 5 bad.\r<BR>\n" );
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document.write( "* **Probability Calculation:** Each of the scenarios for C has a 1/2 probability.<BR>\n" );
document.write( "    * C has 6 bad messages (Scenario 1): Probability = 1/2<BR>\n" );
document.write( "    * C has 5 bad messages (Scenario 2): Probability = 1/2\r<BR>\n" );
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document.write( "* **C has *less* bad messages than A (6):**  This only occurs when C has 5 bad messages.<BR>\n" );
document.write( "* **Probability (C < A) = 1/2 = 0.50**<BR>\n" );
document.write( " </SPAN>\n" );
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