document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1179602>Question 1179602</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If 35% of the people in a community use the emergency room  at a hospital one year. Find these probabilities for a sample of 12 people A) exactly 4 used the emergency room B) At most 4 used the emergency room C) At least 10 used the emergency room  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850207 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=850207\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This problem can be solved using the binomial probability distribution.\r<BR>\n" );
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document.write( "**Given:**\r<BR>\n" );
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document.write( "* Probability of success (using the emergency room): p = 0.35<BR>\n" );
document.write( "* Probability of failure (not using the emergency room): q = 1 - p = 1 - 0.35 = 0.65<BR>\n" );
document.write( "* Sample size: n = 12\r<BR>\n" );
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document.write( "**Binomial Probability Formula:**\r<BR>\n" );
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document.write( "P(X = k) = (nCk) * p^k * q^(n-k)\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "* nCk = n! / (k! * (n-k)!) (the number of combinations of n items taken k at a time)<BR>\n" );
document.write( "* k = number of successes<BR>\n" );
document.write( "* n = sample size\r<BR>\n" );
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document.write( "**A) Exactly 4 used the emergency room (P(X = 4))**\r<BR>\n" );
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document.write( "* P(X = 4) = (12C4) * (0.35)^4 * (0.65)^8<BR>\n" );
document.write( "* 12C4 = 12! / (4! * 8!) = 495<BR>\n" );
document.write( "* P(X = 4) = 495 * (0.01500625) * (0.031864089)<BR>\n" );
document.write( "* P(X = 4) ≈ 0.2366\r<BR>\n" );
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document.write( "**B) At most 4 used the emergency room (P(X ≤ 4))**\r<BR>\n" );
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document.write( "This means we need to find P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4).\r<BR>\n" );
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document.write( "* P(X = 0) = (12C0) * (0.35)^0 * (0.65)^12 ≈ 0.0037<BR>\n" );
document.write( "* P(X = 1) = (12C1) * (0.35)^1 * (0.65)^11 ≈ 0.0261<BR>\n" );
document.write( "* P(X = 2) = (12C2) * (0.35)^2 * (0.65)^10 ≈ 0.0844<BR>\n" );
document.write( "* P(X = 3) = (12C3) * (0.35)^3 * (0.65)^9 ≈ 0.1744<BR>\n" );
document.write( "* P(X = 4) ≈ 0.2366 (calculated in part A)\r<BR>\n" );
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document.write( "* P(X ≤ 4) ≈ 0.0037 + 0.0261 + 0.0844 + 0.1744 + 0.2366 ≈ 0.5252\r<BR>\n" );
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document.write( "**C) At least 10 used the emergency room (P(X ≥ 10))**\r<BR>\n" );
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document.write( "This means we need to find P(X = 10) + P(X = 11) + P(X = 12).\r<BR>\n" );
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document.write( "* P(X = 10) = (12C10) * (0.35)^10 * (0.65)^2 ≈ 0.00018<BR>\n" );
document.write( "* P(X = 11) = (12C11) * (0.35)^11 * (0.65)^1 ≈ 0.000013<BR>\n" );
document.write( "* P(X = 12) = (12C12) * (0.35)^12 * (0.65)^0 ≈ 0.0000003\r<BR>\n" );
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document.write( "* P(X ≥ 10) ≈ 0.00018 + 0.000013 + 0.0000003 ≈ 0.0001933\r<BR>\n" );
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document.write( "**Summary:**\r<BR>\n" );
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document.write( "* A) P(X = 4) ≈ 0.2366<BR>\n" );
document.write( "* B) P(X ≤ 4) ≈ 0.5252<BR>\n" );
document.write( "* C) P(X ≥ 10) ≈ 0.0001933<BR>\n" );
document.write( " </SPAN>\n" );
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