document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200213>Question 1200213</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Deck X has fifteen cards of which 8 are red and 7 are black. Deck Y has twelve cards of which 6 are <BR>\n" );
document.write( "red and 6 are black. A card is drawn at random from Deck X and is discarded. Then all the <BR>\n" );
document.write( "remaining cards are transferred to Deck Y. A card is now drawn from Deck Y and is found to be <BR>\n" );
document.write( "red.\r<BR>\n" );
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document.write( "For this problem, draw a Venn diagram and fill in each of the 4 regions with their correct <BR>\n" );
document.write( "symbols and probabilities using fractions. Show complete solutions. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848139 by </B><A HREF=/tutors/aboutme.mpl?userid=textot><B>textot(100)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=textot><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848139\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Certainly, let's solve this problem using a Venn diagram.\r<BR>\n" );
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document.write( "**1. Define Events:**\r<BR>\n" );
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document.write( "* **R1:** Event of drawing a red card from Deck X.<BR>\n" );
document.write( "* **B1:** Event of drawing a black card from Deck X.<BR>\n" );
document.write( "* **R2:** Event of drawing a red card from Deck Y (after the transfer).<BR>\n" );
document.write( "* **B2:** Event of drawing a black card from Deck Y (after the transfer).\r<BR>\n" );
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document.write( "**2. Calculate Initial Probabilities:**\r<BR>\n" );
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document.write( "* **P(R1) = 8/15** (Probability of drawing a red card from Deck X initially)<BR>\n" );
document.write( "* **P(B1) = 7/15** (Probability of drawing a black card from Deck X initially)\r<BR>\n" );
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document.write( "**3. Determine Probabilities after Transferring Cards:**\r<BR>\n" );
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document.write( "* If a red card is drawn from Deck X (R1):<BR>\n" );
document.write( "    * Deck Y will have 7 red cards and 6 black cards.<BR>\n" );
document.write( "    * P(R2 | R1) = 7/13 <BR>\n" );
document.write( "    * P(B2 | R1) = 6/13\r<BR>\n" );
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document.write( "* If a black card is drawn from Deck X (B1):<BR>\n" );
document.write( "    * Deck Y will have 8 red cards and 5 black cards.<BR>\n" );
document.write( "    * P(R2 | B1) = 8/13<BR>\n" );
document.write( "    * P(B2 | B1) = 5/13\r<BR>\n" );
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document.write( "**4. Calculate Probabilities Using Total Probability Theorem:**\r<BR>\n" );
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document.write( "* P(R2) = P(R1) * P(R2 | R1) + P(B1) * P(R2 | B1)<BR>\n" );
document.write( "* P(R2) = (8/15) * (7/13) + (7/15) * (8/13)<BR>\n" );
document.write( "* P(R2) = 56/195 + 56/195<BR>\n" );
document.write( "* P(R2) = 112/195\r<BR>\n" );
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document.write( "**5. Create the Venn Diagram:**\r<BR>\n" );
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document.write( "* **Region 1 (R2 ∩ R1):** <BR>\n" );
document.write( "    * Represents the event of drawing a red card from Deck X (R1) and then a red card from Deck Y (R2).<BR>\n" );
document.write( "    * Probability: P(R1) * P(R2 | R1) = (8/15) * (7/13) = 56/195\r<BR>\n" );
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document.write( "* **Region 2 (R2 ∩ B1):** <BR>\n" );
document.write( "    * Represents the event of drawing a black card from Deck X (B1) and then a red card from Deck Y (R2).<BR>\n" );
document.write( "    * Probability: P(B1) * P(R2 | B1) = (7/15) * (8/13) = 56/195\r<BR>\n" );
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document.write( "* **Region 3 (B2 ∩ R1):** <BR>\n" );
document.write( "    * Represents the event of drawing a red card from Deck X (R1) and then a black card from Deck Y (B2).<BR>\n" );
document.write( "    * Probability: P(R1) * P(B2 | R1) = (8/15) * (6/13) = 48/195\r<BR>\n" );
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document.write( "* **Region 4 (B2 ∩ B1):** <BR>\n" );
document.write( "    * Represents the event of drawing a black card from Deck X (B1) and then a black card from Deck Y (B2).<BR>\n" );
document.write( "    * Probability: P(B1) * P(B2 | B1) = (7/15) * (5/13) = 35/195\r<BR>\n" );
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document.write( "**Venn Diagram:**\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "     R2            B2<BR>\n" );
document.write( "      |              |<BR>\n" );
document.write( "  -----|--------------|-----<BR>\n" );
document.write( "  | 56/195 | 48/195 |<BR>\n" );
document.write( "  -----|--------------|-----<BR>\n" );
document.write( "  | 56/195 | 35/195 |<BR>\n" );
document.write( "      |              |<BR>\n" );
document.write( "     R1            B1 <BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "**Therefore, the Venn diagram with the probabilities for each region is as shown above.**\r<BR>\n" );
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document.write( "Let me know if you have any other questions or would like to explore further!<BR>\n" );
document.write( " </SPAN>\n" );
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