document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209733>Question 1209733</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 60% of all violent felons in the prison system are repeat offenders. If 45 violent felons are randomly selected, find the probability that  \r<BR>\n" );
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document.write( "a. Exactly 26 of them are repeat offenders. \r<BR>\n" );
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document.write( "b. At most 29 of them are repeat offenders. \r<BR>\n" );
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document.write( "c. At least 29 of them are repeat offenders. \r<BR>\n" );
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document.write( "d. Between 25 and 33 (including 25 and 33) of them are repeat offenders. \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 #850051 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=850051\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this binomial probability problem:\r<BR>\n" );
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document.write( "* **n** (number of trials) = 45<BR>\n" );
document.write( "* **p** (probability of success - repeat offender) = 0.60<BR>\n" );
document.write( "* **q** (probability of failure - not a repeat offender) = 1 - p = 0.40\r<BR>\n" );
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document.write( "The binomial probability formula is: P(x) = (nCx) * p^x * q^(n-x)\r<BR>\n" );
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document.write( "Where nCx represents \"n choose x\" (the binomial coefficient).\r<BR>\n" );
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document.write( "**(a) Exactly 26 are repeat offenders:**\r<BR>\n" );
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document.write( "P(x = 26) = (45C26) * (0.60)^26 * (0.40)^19<BR>\n" );
document.write( "P(x = 26) ≈ 0.0745\r<BR>\n" );
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document.write( "**(b) At most 29 are repeat offenders:**\r<BR>\n" );
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document.write( "This means 0 to 29 are repeat offenders. This is a cumulative probability; we need P(x ≤ 29).  It's best to use a binomial cumulative distribution function (CDF) calculator or statistical software for this.\r<BR>\n" );
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document.write( "P(x ≤ 29) ≈ 0.6617\r<BR>\n" );
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document.write( "**(c) At least 29 are repeat offenders:**\r<BR>\n" );
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document.write( "This means 29 to 45 are repeat offenders. We can use the complement rule:\r<BR>\n" );
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document.write( "P(x ≥ 29) = 1 - P(x < 29) = 1 - P(x ≤ 28)\r<BR>\n" );
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document.write( "Use a binomial CDF calculator:<BR>\n" );
document.write( "P(x ≥ 29) = 1 - 0.5521<BR>\n" );
document.write( "P(x ≥ 29) ≈ 0.4479\r<BR>\n" );
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document.write( "**(d) Between 25 and 33 (inclusive):**\r<BR>\n" );
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document.write( "This means 25, 26, 27, 28, 29, 30, 31, 32, and 33 are repeat offenders. We can use the CDF:\r<BR>\n" );
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document.write( "P(25 ≤ x ≤ 33) = P(x ≤ 33) - P(x ≤ 24)\r<BR>\n" );
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document.write( "Use a binomial CDF calculator:<BR>\n" );
document.write( "P(25 ≤ x ≤ 33) = 0.9272 - 0.1837<BR>\n" );
document.write( "P(25 ≤ x ≤ 33) ≈ 0.7435\r<BR>\n" );
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document.write( "**Summary of Answers:**\r<BR>\n" );
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document.write( "* (a) P(x = 26) ≈ 0.0745<BR>\n" );
document.write( "* (b) P(x ≤ 29) ≈ 0.6617<BR>\n" );
document.write( "* (c) P(x ≥ 29) ≈ 0.4479<BR>\n" );
document.write( "* (d) P(25 ≤ x ≤ 33) ≈ 0.7435<BR>\n" );
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