document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1185670>Question 1185670</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Medical billing errors and fraud are on the rise. According to the MBAA website, 8 out of 10 times, the medical bills that you get are not right. If a sample of 10 medical bills is selected, what is the probability that<BR>\n" );
document.write( "a. 0 medical bills will contain errors?<BR>\n" );
document.write( "b. Exactly 5 medical bills will contain errors?<BR>\n" );
document.write( "c. More than 5 medical bills will contain errors?<BR>\n" );
document.write( "d. What are the mean and standard deviation of the probability distribution </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849675 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=849675\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this problem using the binomial probability distribution:\r<BR>\n" );
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document.write( "**Understanding the Problem**\r<BR>\n" );
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document.write( "This is a binomial probability problem because:\r<BR>\n" );
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document.write( "* There are a fixed number of trials (n = 10 medical bills).<BR>\n" );
document.write( "* Each trial is independent.<BR>\n" );
document.write( "* There are only two outcomes (the bill contains errors or it doesn't).<BR>\n" );
document.write( "* The probability of success (a bill containing errors) is constant (p = 0.8).<BR>\n" );
document.write( "* The probability of failure (a bill not containing errors) is q = 1 - p = 0.2.\r<BR>\n" );
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document.write( "The binomial probability formula is:\r<BR>\n" );
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document.write( "P(x) = (nCx) * p^x * q^(n-x)\r<BR>\n" );
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document.write( "where:\r<BR>\n" );
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document.write( "* P(x) is the probability of x successes<BR>\n" );
document.write( "* n is the number of trials<BR>\n" );
document.write( "* x is the number of successes<BR>\n" );
document.write( "* p is the probability of success<BR>\n" );
document.write( "* q is the probability of failure<BR>\n" );
document.write( "* nCx is the binomial coefficient, calculated as n! / (x! * (n-x)!)\r<BR>\n" );
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document.write( "**a. 0 medical bills will contain errors:**\r<BR>\n" );
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document.write( "P(0) = (10C0) * (0.8)^0 * (0.2)^10<BR>\n" );
document.write( "P(0) = 1 * 1 * (0.2)^10<BR>\n" );
document.write( "P(0) ≈ 1.024 x 10^-7\r<BR>\n" );
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document.write( "**b. Exactly 5 medical bills will contain errors:**\r<BR>\n" );
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document.write( "P(5) = (10C5) * (0.8)^5 * (0.2)^5<BR>\n" );
document.write( "P(5) = 252 * 0.32768 * 0.00032<BR>\n" );
document.write( "P(5) ≈ 0.0264\r<BR>\n" );
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document.write( "**c. More than 5 medical bills will contain errors:**\r<BR>\n" );
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document.write( "This means we need to find the probability of 6, 7, 8, 9, or 10 bills containing errors and add them together. It's often easier to use the complement rule:\r<BR>\n" );
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document.write( "P(x > 5) = 1 - [P(0) + P(1) + P(2) + P(3) + P(4) + P(5)]\r<BR>\n" );
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document.write( "Calculating each of these probabilities and summing them, then subtracting from 1, will give you P(x > 5).  Due to the small probability of P(0) and P(1), we can ignore the results of P(0), P(1)\r<BR>\n" );
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document.write( "P(2) = (10C2) * (0.8)^2 * (0.2)^8 ≈ 0.00088<BR>\n" );
document.write( "P(3) = (10C3) * (0.8)^3 * (0.2)^7 ≈ 0.0088<BR>\n" );
document.write( "P(4) = (10C4) * (0.8)^4 * (0.2)^6 ≈ 0.088<BR>\n" );
document.write( "P(5) ≈ 0.264\r<BR>\n" );
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document.write( "P(x > 5) = 1 - [0.00088 + 0.0088 + 0.088 + 0.264] ≈ 1 - 0.36168 ≈ 0.638\r<BR>\n" );
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document.write( "**d. Mean and Standard Deviation:**\r<BR>\n" );
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document.write( "For a binomial distribution:\r<BR>\n" );
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document.write( "* Mean (μ) = n * p = 10 * 0.8 = 8<BR>\n" );
document.write( "* Standard deviation (σ) = sqrt(n * p * q) = sqrt(10 * 0.8 * 0.2) = sqrt(1.6) ≈ 1.265<BR>\n" );
document.write( " </SPAN>\n" );
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