document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=578837>Question 578837</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 5.	You have two biased coins. Coin A comes up heads with probabaility ¼. Coin B comes up heads with probability ¾. However, you are not sure which is which so you choose a coin randomly and you flip it. If the flip is heads, you guess that the flipped coin is B; othervise you guess that the flipped coin is A. Let events A and B designate which coin was picked. What is the probability P(C) that your guess is correct? (Hint: This is an example of sequential experiments, so you might want to use tree diagrams.)\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #370886 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=370886\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We know that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE P(head | B) = \frac{3}{4}\">. We want to find <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE P(B |head)\">. We use Bayes' theorem:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE P(B|head) = P(head|B)\frac{P(B)}{P(head)}\">.\r<BR>\n" );
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document.write( "We know that P(B) = 1/2 because coins A and B are equally likely to be picked. P(head) is also 1/2 because\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE P(head) = P(A)P(head|A) + P(B)P(head|B)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE = \frac{1}{2}(\frac{1}{4}) + \frac{1}{2}(\frac{3}{4}) = \frac{1}{2}\">\r<BR>\n" );
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document.write( "Therefore,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE P(B|head) = \frac{3}{4}\">\r<BR>\n" );
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document.write( "Additionally, P(A|tail) = 3/4. The answer is *not* 3/4 + 3/4 = 3/2, that's absurd. It is 3/4 because P(head) and P(tail) both have a 1/2 chance of occurring, and we need to multiply by 1/2 to compensate for each cases. </SPAN>\n" );
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