document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1162634>Question 1162634</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An analyst estimates that the probability of default on a seven-year AA-rated bond is 0.47, while that on a seven-year A-rated bond is 0.53. The probability that they will both default is 0.45.\r<BR>\n" );
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document.write( "a. What is the probability that at least one of the bonds defaults? (Round your answer to 2 decimal places.)\r<BR>\n" );
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document.write( "b. What is the probability that neither the seven-year AA-rated bond nor the seven-year A-rated bond defaults? (Round your answer to 2 decimal places.)\r<BR>\n" );
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document.write( "c. Given that the seven-year AA-rated bond defaults, what is the probability that the seven-year A-rated bond also defaults? (Round your answer to 2 decimal places.)\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854398 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=854398\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=2<BR>\n" );
document.write( "p_aa = 0.47<BR>\n" );
document.write( "p_a = 0.53<BR>\n" );
document.write( "p_both = 0.45\r<BR>\n" );
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document.write( "# a. At least one defaults<BR>\n" );
document.write( "p_at_least_one = p_aa + p_a - p_both\r<BR>\n" );
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document.write( "# b. Neither defaults<BR>\n" );
document.write( "p_neither = 1 - p_at_least_one\r<BR>\n" );
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document.write( "# c. P(A defaults | AA defaults)<BR>\n" );
document.write( "p_a_given_aa = p_both / p_aa\r<BR>\n" );
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document.write( "print(f\"{p_at_least_one=}\")<BR>\n" );
document.write( "print(f\"{p_neither=}\")<BR>\n" );
document.write( "print(f\"{p_a_given_aa=}\")\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "p_at_least_one=0.55<BR>\n" );
document.write( "p_neither=0.44999999999999996<BR>\n" );
document.write( "p_a_given_aa=0.9574468085106383\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "To find the requested probabilities, we define the following events:<BR>\n" );
document.write( "* $P(AA) = 0.47$: Probability that the seven-year AA-rated bond defaults.<BR>\n" );
document.write( "* $P(A) = 0.53$: Probability that the seven-year A-rated bond defaults.<BR>\n" );
document.write( "* $P(AA \cap A) = 0.45$: Probability that both bonds default.\r<BR>\n" );
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document.write( "### a. Probability that at least one of the bonds defaults<BR>\n" );
document.write( "The probability that at least one bond defaults is represented by the union of the two events, $P(AA \cup A)$. We use the Addition Rule for Probability:<BR>\n" );
document.write( "$$P(AA \cup A) = P(AA) + P(A) - P(AA \cap A)$$<BR>\n" );
document.write( "$$P(AA \cup A) = 0.47 + 0.53 - 0.45$$<BR>\n" );
document.write( "$$P(AA \cup A) = 1.00 - 0.45 = 0.55$$\r<BR>\n" );
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document.write( "**Probability that at least one defaults: 0.55**\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### b. Probability that neither bond defaults<BR>\n" );
document.write( "The event that neither bond defaults is the complement of the event that at least one defaults.<BR>\n" );
document.write( "$$P(\text{neither}) = 1 - P(AA \cup A)$$<BR>\n" );
document.write( "$$P(\text{neither}) = 1 - 0.55 = 0.45$$\r<BR>\n" );
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document.write( "**Probability that neither defaults: 0.45**\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### c. Probability that the A-rated bond defaults given the AA-rated bond defaults<BR>\n" );
document.write( "This is a conditional probability, denoted as $P(A | AA)$. The formula for conditional probability is:<BR>\n" );
document.write( "$$P(A | AA) = \frac{P(A \cap AA)}{P(AA)}$$<BR>\n" );
document.write( "$$P(A | AA) = \frac{0.45}{0.47} \approx 0.9574$$\r<BR>\n" );
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document.write( "Rounding to 2 decimal places:<BR>\n" );
document.write( "**Probability: 0.96** </SPAN>\n" );
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