document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206345>Question 1206345</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Thirty-five rose bushes are available from a nursery containing 16 red bushes, 11 yellow bushes, and 8 white bushes. If 10 rose bushes are randomly chosen with replacement, then what is the probability that 7 or more are yellow?\r<BR>\n" );
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document.write( "0.0587<BR>\n" );
document.write( "0.0139<BR>\n" );
document.write( "0.1171<BR>\n" );
document.write( "0.9978<BR>\n" );
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document.write( "Answer:  <font color=red size=4>0.0139</font>\r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "We use the binomial probability distribution.<BR>\n" );
document.write( "n = 10 trials<BR>\n" );
document.write( "p = 11/35 = probability of getting a yellow rose on any single trial<BR>\n" );
document.write( "x = number of yellow roses selected.\r<BR>\n" );
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document.write( "x will be selected from the set {7,8,9,10}\r<BR>\n" );
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document.write( "The binomial probability formula is<BR>\n" );
document.write( "B(x) = (nCx)*(p)^x*(1-p)^(n-x)<BR>\n" );
document.write( "where the nCx refers to the nCr combination formula.<BR>\n" );
document.write( "To compute the nCr value in a spreadsheet, use the command called <a  rel=nofollow HREF=\"https://support.microsoft.com/en-us/office/combin-function-12a3f276-0a21-423a-8de6-06990aaf638a\">Combin</a><BR>\n" );
document.write( "For example, type in <font color=blue>=Combin(10,7)</font> to mean 10C7.<BR>\n" );
document.write( "Do not use the \"choose\" command. That's something entirely different.\r<BR>\n" );
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document.write( "Let's find the probability of selecting exactly 7 yellow roses and 10-7 = 3 nonyellow roses.<BR>\n" );
document.write( "B(x) = (nCx)*(p)^x*(1-p)^(n-x)<BR>\n" );
document.write( "B(x) = (10Cx)*(11/35)^x*(1-11/35)^(10-x)<BR>\n" );
document.write( "B(x) = (10Cx)*(11/35)^x*(24/35)^(10-x)<BR>\n" );
document.write( "B(7) = (10C7)*(11/35)^7*(24/35)^(10-7)<BR>\n" );
document.write( "B(7) = (120)*(11/35)^7*(24/35)^3<BR>\n" );
document.write( "B(7) = 0.0117188 approximately\r<BR>\n" );
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document.write( "Repeat similar calculations for x = 8 through x = 10.<BR>\n" );
document.write( "This is what the table of values should look like<BR>\n" );
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document.write( "<tr><TD >x</TD><TD >B(x)</TD></tr><tr><TD >7</TD><TD >0.0117188</TD></tr><tr><TD >8</TD><TD >0.00201417</TD></tr><tr><TD >9</TD><TD >0.00020515</TD></tr><tr><TD >10</TD><TD >0.0000094</TD></tr>\n" );
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document.write( "Add up the P(x) values to get:<BR>\n" );
document.write( "0.0117188 + 0.00201417 + 0.00020515 + 0.0000094 = 0.01394752 which rounds to <font color=red size=4>0.0139</font>\r<BR>\n" );
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document.write( "A shortcut would be to use the <a  rel=nofollow HREF=\"https://support.google.com/docs/answer/3093987?hl=en\">BinomDist</a> function on a spreadsheet.<BR>\n" );
document.write( "The one line calculation to type in is <font color=blue>=1-BinomDist(6,10,11/35,1)</font><BR>\n" );
document.write( "The BinomDist(6,10,11/35,1) portion calculates the binomial cdf value from x = 0 to x = 6. Subtracting that result from 1 leads to B(7)+B(8)+B(9)+B(10)\r<BR>\n" );
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document.write( "Note:<BR>\n" );
document.write( "B(0)+B(1)+B(2)+B(3)+B(4)+B(5)+B(6)+<font color=blue>B(7)+B(8)+B(9)+B(10)</font> = 1<BR>\n" );
document.write( "<font color=blue>B(7)+B(8)+B(9)+B(10)</font> = 1 - (B(0)+B(1)+B(2)+B(3)+B(4)+B(5)+B(6))\r<BR>\n" );
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document.write( "If you want to round the result then you can type in <font color=blue>=Round(1-BinomDist(6,10,11/35,1),4)</font><BR>\n" );
document.write( "Or you can place the original calculation in cell A1 and then type in <font color=blue>=Round(A1,4)</font> so you round the contents of A1 to 4 decimal places.<BR>\n" );
document.write( "Change A1 to whatever other cell you want.\r<BR>\n" );
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document.write( "If you are using a TI84, then refer to <a  rel=nofollow HREF=\"https://www.statology.org/binomial-probabilities-ti-84-calculator/\">this article</a><BR>\n" );
document.write( "Use binomCDF instead of binomPDF.<BR>\n" );
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