document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=473880>Question 473880</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can someone help with this problem, I don't have a calculator:\r<BR>\n" );
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document.write( "43% of adults say cashews are their favorite kind of nut.  You randomly select 12 adults and ask each to name his or her favorite nut. Find the probability that the number who say cashews are their favorite is a)exactly three, (b) at least four and (c) at most two.  If convenient use technology to find the probabilities.<BR>\n" );
document.write( "(a) P(3)=    (round to the nearest thousandth as needed)<BR>\n" );
document.write( "(b) P(x&#8805;4)=  (round to the nearest thousandth as needed)<BR>\n" );
document.write( "(c) P(&#8804;2)= (round to the nearest thousandth as needed)<BR>\n" );
document.write( "Thank you! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #325035 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=325035\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> All of these problems involve the binomial distribution. To verify your work that deals with the binomial distribution, check out <a  rel=nofollow HREF=\"http://stattrek.com/tables/binomial.aspx\">this calculator</a>.\r<BR>\n" );
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document.write( "a)\r<BR>\n" );
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document.write( "P(X = 3) = (12 C 3)*(0.43)^(3)*(1-0.43)^(12-3) \r<BR>\n" );
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document.write( "P(X = 3) = (12 C 3)*(0.43)^(3)*(0.57)^(12-3) \r<BR>\n" );
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document.write( "Note: 12 C 3 = (12!)/(3!(12-3)!) = 220\r<BR>\n" );
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document.write( "P(X = 3) = (220)*(0.43)^(3)*(0.57)^9 \r<BR>\n" );
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document.write( "P(X = 3) = (220)*(0.079507)*(0.006351461955384057) \r<BR>\n" );
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document.write( "P(X = 3) = 0.11109685085107844837778\r<BR>\n" );
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document.write( "P(X = 3) = 0.111\r<BR>\n" );
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document.write( "So the answer is 0.111\r<BR>\n" );
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document.write( "b)\r<BR>\n" );
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document.write( "To calculate P(X >= 4), it's easier to first calculate P(X < 4) and then subtract this probability from 1. So to find P(X < 4), add up the probabilities for X=0 up to X=3 (we're not including 4)\r<BR>\n" );
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document.write( "P(X = 0) = (12 C 0)*(0.43)^(0)*(1-0.43)^(12-0) = (12 C 0)*(0.43)^(0)*(0.57)^(12-0) = (1)*(0.43)^(0)*(0.57)^12 = (1)*(1)*(0.001176246293903439668001) = 0.001176246293903439668001\r<BR>\n" );
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document.write( "P(X = 1) = (12 C 1)*(0.43)^(1)*(1-0.43)^(12-1) = (12 C 1)*(0.43)^(1)*(0.57)^(12-1) = (12)*(0.43)^(1)*(0.57)^11 = (12)*(0.43)*(0.0020635899893042801193) = 0.010648124344810085415588\r<BR>\n" );
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document.write( "P(X = 2) = (12 C 2)*(0.43)^(2)*(1-0.43)^(12-2) = (12 C 2)*(0.43)^(2)*(0.57)^(12-2) = (66)*(0.43)^(2)*(0.57)^10 = (66)*(0.1849)*(0.00362033331456891249) = 0.044180375571010266680466\r<BR>\n" );
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document.write( "P(X = 3) = (12 ncr 3)*(0.43)^(3)*(1-0.43)^(12-3) = (12 ncr 3)*(0.43)^(3)*(0.57)^(12-3) = (220)*(0.43)^(3)*(0.57)^9 = (220)*(0.079507)*(0.006351461955384057) = 0.11109685085107844837778\r<BR>\n" );
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document.write( "So \r<BR>\n" );
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document.write( "P(X = 0)= 0.001176246293903439668001\r<BR>\n" );
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document.write( "P(X = 1)= 0.010648124344810085415588\r<BR>\n" );
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document.write( "P(X = 2)= 0.044180375571010266680466\r<BR>\n" );
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document.write( "P(X = 3)= 0.11109685085107844837778\r<BR>\n" );
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document.write( "Now onto the main event...\r<BR>\n" );
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document.write( "P(X >= 4) = 1 - P(X < 4)\r<BR>\n" );
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document.write( "P(X >= 4) = 1- [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)]\r<BR>\n" );
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document.write( "P(X >= 4) = 1- [ 0.001176246293903439668001 + 0.010648124344810085415588 + 0.044180375571010266680466 + 0.11109685085107844837778]\r<BR>\n" );
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document.write( "P(X >= 4) = 1-0.1671015970608\r<BR>\n" );
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document.write( "P(X >= 4) = 0.8328984029392\r<BR>\n" );
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document.write( "P(X >= 4) = 0.833\r<BR>\n" );
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document.write( "So the answer is 0.833\r<BR>\n" );
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document.write( "c)\r<BR>\n" );
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document.write( "To calculate P(X <= 2), simply add up all the probabilities from X=0 to X=2. So you'll need to calculate the following first\r<BR>\n" );
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document.write( "P(X = 0) = (12 C 0)*(0.43)^(0)*(1-0.43)^(12-0) = (12 C 0)*(0.43)^(0)*(0.57)^(12-0) = (1)*(0.43)^(0)*(0.57)^12 = (1)*(1)*(0.001176246293903439668001) = 0.001176246293903439668001\r<BR>\n" );
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document.write( "P(X = 1) = (12 C 1)*(0.43)^(1)*(1-0.43)^(12-1) = (12 C 1)*(0.43)^(1)*(0.57)^(12-1) = (12)*(0.43)^(1)*(0.57)^11 = (12)*(0.43)*(0.0020635899893042801193) = 0.010648124344810085415588\r<BR>\n" );
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document.write( "P(X = 2) = (12 C 2)*(0.43)^(2)*(1-0.43)^(12-2) = (12 C 2)*(0.43)^(2)*(0.57)^(12-2) = (66)*(0.43)^(2)*(0.57)^10 = (66)*(0.1849)*(0.00362033331456891249) = 0.044180375571010266680466\r<BR>\n" );
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document.write( "So \r<BR>\n" );
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document.write( "P(X = 0)= 0.001176246293903439668001\r<BR>\n" );
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document.write( "P(X = 1)= 0.010648124344810085415588\r<BR>\n" );
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document.write( "P(X = 2)= 0.044180375571010266680466\r<BR>\n" );
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document.write( "Now onto the main event...\r<BR>\n" );
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document.write( "P(X <= 2) = P(X = 0) + P(X = 1) + P(X = 2) \r<BR>\n" );
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document.write( "P(X <= 2) = 0.001176246293903439668001 + 0.010648124344810085415588 + 0.044180375571010266680466 \r<BR>\n" );
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document.write( "P(X <= 2) = 0.056004746209723791764055\r<BR>\n" );
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document.write( "P(X <= 2) = 0.056\r<BR>\n" );
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document.write( "So the answer is 0.056\r<BR>\n" );
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document.write( "To verify other work that deals with the binomial distribution, check out <a  rel=nofollow HREF=\"http://stattrek.com/tables/binomial.aspx\">this calculator</a>. </SPAN>\n" );
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