document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192908>Question 1192908</A>: <I><SPAN STYLE=\"word-break: break-all;\"> There are 267 students in faculty and each of them spends approximately 5.625 hours daily in the reading room. Reading room works 8 hours a day. How many seats must be in the reading room that the probability of finding a free seat is not less than 0.78? Use the De Moivre  -  Laplace theorem </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848947 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=848947\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Determine the total student-hours in the reading room:**\r<BR>\n" );
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document.write( "* Total student-hours per day = Number of students * Hours per student per day<BR>\n" );
document.write( "* Total student-hours per day = 267 students * 5.625 hours/student = 1503.75 student-hours\r<BR>\n" );
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document.write( "**2. Calculate the average student-hours per hour of reading room operation:**\r<BR>\n" );
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document.write( "* Average student-hours per hour = Total student-hours per day / Reading room hours per day<BR>\n" );
document.write( "* Average student-hours per hour = 1503.75 student-hours / 8 hours = 187.96875 student-hours/hour\r<BR>\n" );
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document.write( "**3. Determine the minimum number of seats required:**\r<BR>\n" );
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document.write( "* We need to find the minimum number of seats (S) such that the probability of finding a free seat is at least 0.78. <BR>\n" );
document.write( "* This means the probability of all seats being occupied is at most 1 - 0.78 = 0.22.\r<BR>\n" );
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document.write( "* **Using the De Moivre-Laplace Theorem:**\r<BR>\n" );
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document.write( "    * The De Moivre-Laplace theorem states that for a large number of independent trials (in this case, the number of students using the reading room at any given hour), the binomial distribution can be approximated by a normal distribution.\r<BR>\n" );
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document.write( "    * We can model the number of students using the reading room at any given hour as a random variable with a binomial distribution. \r<BR>\n" );
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document.write( "    * Let:<BR>\n" );
document.write( "        * X = the number of students using the reading room at a given hour<BR>\n" );
document.write( "        * n = the number of students (267)<BR>\n" );
document.write( "        * p = the probability that a student is using the reading room at any given hour (p = 5.625 hours/student / 8 hours/day = 0.703125)<BR>\n" );
document.write( "        * μ = mean of X = n * p = 267 * 0.703125 = 187.96875 <BR>\n" );
document.write( "        * σ = standard deviation of X = √(n * p * (1 - p)) = √(267 * 0.703125 * 0.296875) ≈ 7.44\r<BR>\n" );
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document.write( "    * We want to find the minimum number of seats (S) such that:<BR>\n" );
document.write( "        * P(X > S) ≤ 0.22\r<BR>\n" );
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document.write( "    * **Standardize:**<BR>\n" );
document.write( "        * z = (S - μ) / σ <BR>\n" );
document.write( "        * Find the z-score corresponding to a cumulative probability of 0.78 (since we want P(X > S) ≤ 0.22) in a standard normal distribution table. This z-score is approximately 0.77.\r<BR>\n" );
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document.write( "    * **Solve for S:**<BR>\n" );
document.write( "        * 0.77 = (S - 187.96875) / 7.44<BR>\n" );
document.write( "        * S = 0.77 * 7.44 + 187.96875<BR>\n" );
document.write( "        * S ≈ 193.6\r<BR>\n" );
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document.write( "* **Therefore, the minimum number of seats required in the reading room is approximately 194 to ensure that the probability of finding a free seat is not less than 0.78.**\r<BR>\n" );
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document.write( "**Note:** This calculation uses the De Moivre-Laplace theorem as an approximation. For a more precise calculation, you could use the binomial distribution directly. However, with a large number of students (n = 267), the normal approximation should provide a reasonably accurate result.<BR>\n" );
document.write( " </SPAN>\n" );
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