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 #852009 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52781)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852009\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "60% of all violent felons in the prison system are repeat offenders. <BR>\n" );
document.write( "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.<BR>\n" );
document.write( "b. At most 29 of them are repeat offenders.<BR>\n" );
document.write( "c. At least 29 of them are repeat offenders.<BR>\n" );
document.write( "d. Between 25 and 33 (including 25 and 33) of them are repeat offenders.<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;As it regularly happens with &nbsp;@CPhill' solutions on &nbsp;Binomial distribution,\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;all calculations &nbsp;(a), &nbsp;(b), &nbsp;(c) &nbsp;and &nbsp;(d) &nbsp;in the post by &nbsp;@CPhill are incorrect.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Below is my correct solution.\r<BR>\n" );
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document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "Here's how to solve this binomial probability problem:\r\n" );
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document.write( "* **n** (number of trials) = 45\r\n" );
document.write( "* **p** (probability of success - repeat offender) = 0.60\r\n" );
document.write( "* **q** (probability of failure - not a repeat offender) = 1 - p = 0.40\r\n" );
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document.write( "The binomial probability formula is: P(x) = (nCx) * p^x * q^(n-x)\r\n" );
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document.write( "Where nCx represents \"n choose x\" (the binomial coefficient).\r\n" );
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document.write( "**(a) Exactly 26 are repeat offenders:**\r\n" );
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document.write( "P(x = 26) = (45C26) * (0.60)^26 * (0.40)^19\r\n" );
document.write( "P(x = 26) ≈ 0.1143\r\n" );
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document.write( "**(b) At most 29 are repeat offenders:**\r\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).  \r\n" );
document.write( "It's best to use a binomial cumulative distribution function (CDF) calculator or statistical software for this.\r\n" );
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document.write( "P(x ≤ 29) ≈ 0.7751\r\n" );
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document.write( "**(c) At least 29 are repeat offenders:**\r\n" );
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document.write( "This means 29 to 45 are repeat offenders. We can use the complement rule:\r\n" );
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document.write( "P(x ≥ 29) = 1 - P(x < 29) = 1 - P(x ≤ 28)\r\n" );
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document.write( "Use a binomial CDF calculator:\r\n" );
document.write( "P(x ≥ 29) = 1 - 0.6728\r\n" );
document.write( "P(x ≥ 29) ≈ 0.3272\r\n" );
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document.write( "**(d) Between 25 and 33 (inclusive):**\r\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\n" );
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document.write( "P(25 ≤ x ≤ 33) = P(x ≤ 33) - P(x ≤ 24)\r\n" );
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document.write( "Use a binomial CDF calculator:\r\n" );
document.write( "P(25 ≤ x ≤ 33) = 0.9784 - 0.2223\r\n" );
document.write( "P(25 ≤ x ≤ 33) ≈ 0.7561\r\n" );
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document.write( "**Summary of Answers:**\r\n" );
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document.write( "* (a) P(x = 26) ≈ 0.1143\r\n" );
document.write( "* (b) P(x ≤ 29) ≈ 0.7751\r\n" );
document.write( "* (c) P(x ≥ 29) ≈ 0.3272\r\n" );
document.write( "* (d) P(25 ≤ x ≤ 33) ≈ 0.7561\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "Solved CORRECTLY.\r<BR>\n" );
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document.write( "Practically all calculations by @CPhill to this kind of problems on Binomial distribution are INCORRECT.\r<BR>\n" );
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document.write( "To check your/my calculations, &nbsp;use online calculator \r<BR>\n" );
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document.write( "https://stattrek.com/online-calculator/binomial\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Regarding the post by @CPhill . . . \r<BR>\n" );
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document.write( "Keep in mind that @CPhill is a pseudonym for the Google artificial intelligence.\r<BR>\n" );
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document.write( "The artificial intelligence is like a baby now. It is in the experimental stage <BR>\n" );
document.write( "of development and can make mistakes and produce nonsense without any embarrassment.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It has no feeling of shame - it is shameless.\r<BR>\n" );
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document.write( "This time, again, &nbsp;it made an error.\r<BR>\n" );
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document.write( "Although the @CPhill' solutions are copy-paste &nbsp;Google &nbsp;AI solutions, &nbsp;there is one essential difference.\r<BR>\n" );
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document.write( "Every time, &nbsp;Google &nbsp;AI &nbsp;makes a note at the end of its solutions that &nbsp;Google &nbsp;AI &nbsp;is experimental<BR>\n" );
document.write( "and can make errors/mistakes.\r<BR>\n" );
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document.write( "All @CPhill' solutions are copy-paste of &nbsp;Google &nbsp;AI &nbsp;solutions, with one difference:<BR>\n" );
document.write( "@PChill never makes this notice and never says that his solutions are copy-past that of Google.<BR>\n" );
document.write( "So, &nbsp;he &nbsp;NEVER &nbsp;SAYS &nbsp;TRUTH.\r<BR>\n" );
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document.write( "Every time, &nbsp;@CPhill embarrassed to tell the truth.\r<BR>\n" );
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document.write( "But I am not embarrassing to tell the truth, &nbsp;as it is my duty at this forum.\r<BR>\n" );
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document.write( "And the last my comment.\r<BR>\n" );
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document.write( "When you obtain such posts from @CPhill, &nbsp;remember, &nbsp;that &nbsp;NOBODY &nbsp;is responsible for their correctness, <BR>\n" );
document.write( "until the specialists and experts will check and confirm their correctness.\r<BR>\n" );
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document.write( "Without it, &nbsp;their reliability is &nbsp;ZERO and their creadability is &nbsp;ZERO, &nbsp;too.\r<BR>\n" );
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