document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186891>Question 1186891</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In an ablation procedure, the probability of acute success (determined at completion of <BR>\n" );
document.write( "the procedure) is 0.95 when an image mapping system is used. Without the image <BR>\n" );
document.write( "mapping system, the probably of acute success is only0.80. Suppose that Patient A is <BR>\n" );
document.write( "given the treatment with the mapping system and Patient B is given the treatment <BR>\n" );
document.write( "without the mapping system. Determine the following probabilities:<BR>\n" );
document.write( "a. Both patients A and B had acute successes<BR>\n" );
document.write( "b. A had an acute success but B had an acute failure.<BR>\n" );
document.write( "c. B had an acute success but A had an acute failure.<BR>\n" );
document.write( "d. Both A and B had acute failures.<BR>\n" );
document.write( "e. At least one of the patients had an acute success.<BR>\n" );
document.write( "f. Describe two ways that the result in (e) can be calculated based on the results from <BR>\n" );
document.write( "(a), (b), (c), and (d). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849336 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=849336\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to calculate the probabilities:\r<BR>\n" );
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document.write( "Let's define the events:\r<BR>\n" );
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document.write( "*   A: Patient A has acute success.<BR>\n" );
document.write( "*   B: Patient B has acute success.\r<BR>\n" );
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document.write( "We are given:\r<BR>\n" );
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document.write( "*   P(A) = 0.95 (Probability of success with mapping system)<BR>\n" );
document.write( "*   P(B) = 0.80 (Probability of success without mapping system)\r<BR>\n" );
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document.write( "Since the treatments are independent for each patient, we can multiply the probabilities.\r<BR>\n" );
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document.write( "**a. Both patients A and B had acute successes:**\r<BR>\n" );
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document.write( "P(A and B) = P(A) * P(B) = 0.95 * 0.80 = 0.76\r<BR>\n" );
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document.write( "**b. A had an acute success but B had an acute failure:**\r<BR>\n" );
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document.write( "P(A and not B) = P(A) * P(not B) = 0.95 * (1 - 0.80) = 0.95 * 0.20 = 0.19\r<BR>\n" );
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document.write( "**c. B had an acute success but A had an acute failure:**\r<BR>\n" );
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document.write( "P(not A and B) = P(not A) * P(B) = (1 - 0.95) * 0.80 = 0.05 * 0.80 = 0.04\r<BR>\n" );
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document.write( "**d. Both A and B had acute failures:**\r<BR>\n" );
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document.write( "P(not A and not B) = P(not A) * P(not B) = (1 - 0.95) * (1 - 0.80) = 0.05 * 0.20 = 0.01\r<BR>\n" );
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document.write( "**e. At least one of the patients had an acute success:**\r<BR>\n" );
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document.write( "There are two ways to calculate this:\r<BR>\n" );
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document.write( "*   **Method 1 (Direct):**  This is the complement of *both* having acute failures.\r<BR>\n" );
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document.write( "    P(at least one success) = 1 - P(not A and not B) = 1 - 0.01 = 0.99\r<BR>\n" );
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document.write( "*   **Method 2 (Addition):**  Add the probabilities of the scenarios where at least one has a success.\r<BR>\n" );
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document.write( "    P(at least one success) = P(A and B) + P(A and not B) + P(not A and B) = 0.76 + 0.19 + 0.04 = 0.99\r<BR>\n" );
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document.write( "**f. Two ways to calculate (e):**\r<BR>\n" );
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document.write( "As shown above:\r<BR>\n" );
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document.write( "1.  **Complement Rule:** The probability of at least one success is 1 minus the probability that *neither* patient has a success (calculated in part d).\r<BR>\n" );
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document.write( "2.  **Addition Rule (for mutually exclusive events):** Add the probabilities of all the scenarios where at least one patient has a success (calculated in parts a, b, and c).  These scenarios are mutually exclusive, so we can simply add their probabilities.<BR>\n" );
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