document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1082753>Question 1082753</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a particular card game, players receive five cards from a standard deck of 52 cards (consisting of the standard four suits, and 13 cards of each suit). Determine each of the following probabilities:<BR>\n" );
document.write( "a) The probability of 2 or more red cards using a hypergeometric distribution<BR>\n" );
document.write( "b) The probability of 2 or more red cards using a normal approximation. Be sure to check that it is appropriate to use an approximation. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #696874 by </B><A HREF=/tutors/aboutme.mpl?userid=natolino_2017><B>natolino_2017(77)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=natolino_2017><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=696874\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a) P(X=a) = (dCa)((N-d)C(n-a))/(NCn)\r<BR>\n" );
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document.write( "in this case: N = 52 (number of total cards)<BR>\n" );
document.write( "              n = 5 (number of picked cards)<BR>\n" );
document.write( "              d = 26 (number of working cards)\r<BR>\n" );
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document.write( "According to the model, it's easier using the complement of the asked probability.\r<BR>\n" );
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document.write( "P() = 1 - (P(x=1) + P(x=0) <BR>\n" );
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document.write( "      1 - (26C1)(26C4)/(52C5) - (26C2)(26C3)/(52C5) = 5,251/9,996 = 52.531%\r<BR>\n" );
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document.write( "b) First Not that the E(x) = nd/N = 5/2 ( So the expected number of red cards is between of two and three)\r<BR>\n" );
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document.write( "and V(x) = (N-n)(N-d)(nd)/(N^2(N-1)) = 235/204 (So the variation expected is aproximated one between every experiment).\r<BR>\n" );
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document.write( "As N = 52 > 30 it's fair to use a Normal aproximation.\r<BR>\n" );
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document.write( "y is a Normal with u= 2.5 and sigma square = 235/204\r<BR>\n" );
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document.write( "P(y>=2) using standardization P(Z >= (2-2.5)/(sqrt(235/204))\r<BR>\n" );
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document.write( "                              P(Z >= -0.46585) using symmetry.\r<BR>\n" );
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document.write( "                              0.5 + P(0 <= z <= 0.46585) Extrapolating\r<BR>\n" );
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document.write( "                              0.5 + 0.179 = 0.679 = 67.9%\r<BR>\n" );
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document.write( "which is different that the exact value, but the error is less than 30%.\r<BR>\n" );
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document.write( "If we were using several deck of cards the error would minor.\r<BR>\n" );
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document.write( "@natolino_<BR>\n" );
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document.write( " </SPAN>\n" );
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