document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1185553>Question 1185553</A>: <I><SPAN STYLE=\"word-break: break-all;\"> α=14<BR>\n" );
document.write( "Defaulting on a loan means failing to pay it back on time. The default rate among X university<BR>\n" );
document.write( "students on their student loans is α %. As a project you develop a test to predict which students<BR>\n" );
document.write( "will default. Your test is good but not perfect. It gives 4% false positives, i.e. prediciting a<BR>\n" );
document.write( "student will default who in fact will not. If has a 0% false negative rate, i.e. prediciting a student<BR>\n" );
document.write( "won't default who in fact will.<BR>\n" );
document.write( "a) Suppose a random student predicts positive. What is the probability that he will truly<BR>\n" );
document.write( "default.<BR>\n" );
document.write( "b) Someone offers to bet me the student in part(a) won't default. They want me to pay them<BR>\n" );
document.write( "Rs. 5k if the student doesn't default and they'll pay me Rs. 20k if the student does default.<BR>\n" );
document.write( "Is this a good bet for me to take?<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 #849683 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=849683\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to break down this probability problem:\r<BR>\n" );
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document.write( "**a) Probability of True Default Given a Positive Prediction:**\r<BR>\n" );
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document.write( "We'll use Bayes' Theorem to solve this. Let:\r<BR>\n" );
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document.write( "* D be the event that a student defaults.<BR>\n" );
document.write( "* P be the event that the test predicts positive (a student will default).\r<BR>\n" );
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document.write( "We want to find P(D|P), the probability that a student defaults given a positive prediction. Bayes' Theorem states:\r<BR>\n" );
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document.write( "P(D|P) = [P(P|D) * P(D)] / P(P)\r<BR>\n" );
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document.write( "Let's break down the components:\r<BR>\n" );
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document.write( "* **P(D):** This is the default rate, given as α = 14% or 0.14.<BR>\n" );
document.write( "* **P(P|D):** This is the probability of a positive prediction given that the student defaults. Since there are no false negatives, if a student defaults, the test will *always* predict positive. So, P(P|D) = 1.<BR>\n" );
document.write( "* **P(P):** This is the probability of a positive prediction. This can happen in two ways: either the student defaults *and* the test predicts positive (which we already know is just P(D)), or the student *doesn't* default *but* the test predicts positive (a false positive).\r<BR>\n" );
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document.write( "   * Probability of *not* defaulting = 1 - P(D) = 1 - 0.14 = 0.86<BR>\n" );
document.write( "   * Probability of a false positive = 4% = 0.04<BR>\n" );
document.write( "   * So, P(P) = P(D) + [P(not D) * P(false positive)] = 0.14 + (0.86 * 0.04) = 0.14 + 0.0344 = 0.1744\r<BR>\n" );
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document.write( "Now we can plug everything into Bayes' Theorem:\r<BR>\n" );
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document.write( "P(D|P) = (1 * 0.14) / 0.1744 ≈ 0.8026\r<BR>\n" );
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document.write( "Therefore, if the test predicts positive, there's approximately an 80.26% chance that the student will actually default.\r<BR>\n" );
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document.write( "**b) Is the Bet a Good One?**\r<BR>\n" );
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document.write( "To determine if the bet is good, we need to calculate the expected value of the bet.\r<BR>\n" );
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document.write( "* **Outcome 1: Student defaults:** You win Rs. 20,000. The probability of this is P(D|P) ≈ 0.8026.<BR>\n" );
document.write( "* **Outcome 2: Student doesn't default:** You lose Rs. 5,000. The probability of this is 1 - P(D|P) ≈ 1 - 0.8026 = 0.1974.\r<BR>\n" );
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document.write( "Expected Value = (Probability of winning * Amount won) - (Probability of losing * Amount lost)<BR>\n" );
document.write( "Expected Value = (0.8026 * 20000) - (0.1974 * 5000)<BR>\n" );
document.write( "Expected Value = 16052 - 987<BR>\n" );
document.write( "Expected Value = 15065\r<BR>\n" );
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document.write( "Since the expected value is positive (Rs. 15,065), this is a good bet for you to take *in the long run*.  Over many such bets, you would expect to profit.<BR>\n" );
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