document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168481>Question 1168481</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Urn A has 2 white and 13 red balls. Urn B has 14 white and 6 red balls. We flip a fair coin. If the outcome is heads, then a ball from urn A is selected, whereas if the outcome is tails, then a ball from urn B is selected. Suppose that a white ball is selected. What is the probability that the coin landed heads?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851566 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2264)</B></A><img src=\"/images/stars/stars-13.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=851566\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem using Bayes' Theorem.\r<BR>\n" );
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document.write( "**1. Define Events**\r<BR>\n" );
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document.write( "* **H:** Coin lands heads.<BR>\n" );
document.write( "* **T:** Coin lands tails.<BR>\n" );
document.write( "* **W:** A white ball is selected.\r<BR>\n" );
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document.write( "**2. Given Probabilities**\r<BR>\n" );
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document.write( "* P(H) = 1/2 (fair coin)<BR>\n" );
document.write( "* P(T) = 1/2 (fair coin)<BR>\n" );
document.write( "* P(W | H) = 2/15 (probability of white ball from urn A)<BR>\n" );
document.write( "* P(W | T) = 14/20 = 7/10 (probability of white ball from urn B)\r<BR>\n" );
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document.write( "**3. Bayes' Theorem**\r<BR>\n" );
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document.write( "We want to find P(H | W), the probability that the coin landed heads given that a white ball was selected.\r<BR>\n" );
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document.write( "Bayes' Theorem states:\r<BR>\n" );
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document.write( "P(H | W) = [P(W | H) * P(H)] / P(W)\r<BR>\n" );
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document.write( "**4. Calculate P(W)**\r<BR>\n" );
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document.write( "We need to calculate P(W), the probability of selecting a white ball. We can use the law of total probability:\r<BR>\n" );
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document.write( "P(W) = P(W | H) * P(H) + P(W | T) * P(T)\r<BR>\n" );
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document.write( "P(W) = (2/15) * (1/2) + (7/10) * (1/2)\r<BR>\n" );
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document.write( "P(W) = 1/15 + 7/20\r<BR>\n" );
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document.write( "P(W) = (4 + 21) / 60\r<BR>\n" );
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document.write( "P(W) = 25/60 = 5/12\r<BR>\n" );
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document.write( "**5. Calculate P(H | W)**\r<BR>\n" );
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document.write( "Now, we can use Bayes' Theorem:\r<BR>\n" );
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document.write( "P(H | W) = [P(W | H) * P(H)] / P(W)\r<BR>\n" );
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document.write( "P(H | W) = [(2/15) * (1/2)] / (5/12)\r<BR>\n" );
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document.write( "P(H | W) = (1/15) / (5/12)\r<BR>\n" );
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document.write( "P(H | W) = (1/15) * (12/5)\r<BR>\n" );
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document.write( "P(H | W) = 12/75 = 4/25\r<BR>\n" );
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document.write( "**6. Final Answer**\r<BR>\n" );
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document.write( "The probability that the coin landed heads given that a white ball was selected is 4/25.<BR>\n" );
document.write( " </SPAN>\n" );
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