document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201223>Question 1201223</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The manufacturer of a fertilizer guarantees that , with the aid of the fertilizer , 75 % of planted seeds will germinate . Suppose the manufacturer is correct . If 9 seeds planted with the fertilizer are randomly selected , what is the probability that at most 7 of them germinate ? Carry your intermediate computations to at least four decimal places , and round your answer to two decimal places . </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #835511 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=835511\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font color=red>0.70</font>\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "We have a binomial process going on because...<ul><li>The seeds either germinate or they don't. There are two outcomes.</li><li>Each seed's germination is independent of any other (assuming one doesn't cross-contaminate another).</li><li>The probability of germination is the same for any seed.</li></ul>n = 9 seeds total = sample size<BR>\n" );
document.write( "p = 0.75 = probability of germination<BR>\n" );
document.write( "x = number of seeds that germinate.\r<BR>\n" );
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document.write( "The value of x is an integer between 0 and 9 inclusive of each endpoint.<BR>\n" );
document.write( "The value x is some value in the set {0,1,2,3,4,5,6,7,8,9}\r<BR>\n" );
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document.write( "B(x) = binomial probability that exactly x seeds will germinate<BR>\n" );
document.write( "B(x) = (n C x)*(p^x)*(1-p)^(n-x)<BR>\n" );
document.write( "B(x) = (9 C x)*(0.75^x)*(1-0.75)^(9-x)<BR>\n" );
document.write( "B(x) = (9 C x)*(0.75^x)*(0.25)^(9-x)<BR>\n" );
document.write( "The first part 9 C x refers to the nCr combination formula. <BR>\n" );
document.write( "Pascal's Triangle can be used to determine the nCr values.\r<BR>\n" );
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document.write( "We're asked \"what is the probability that <font color=blue>at most 7</font> of them germinate?\"<BR>\n" );
document.write( "Translation: \"What is the probability that <font color=blue>7 or fewer</font> seeds germinate?\"\r<BR>\n" );
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document.write( "We'll need to compute the summation <BR>\n" );
document.write( "B(0)+B(1)+B(2)+B(3)+B(4)+B(5)+B(6)+B(7)<BR>\n" );
document.write( "This is the sum from B(0) to B(7).\r<BR>\n" );
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document.write( "Ok that's quite a lot of things to compute and add up.<BR>\n" );
document.write( "It would be tedious unnecessary busy-work to expect a student to do this (especially by hand or through use of a pocket calculator).\r<BR>\n" );
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document.write( "The much preferred shortcut is to find B(8)+B(9) first. Then subtract that sum from 1.\r<BR>\n" );
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document.write( "This works because:<BR>\n" );
document.write( "<font color=blue>B(0)+B(1)+B(2)+B(3)+B(4)+B(5)+B(6)+B(7)</font>+<font color=red>B(8)+B(9)</font> = 1<BR>\n" );
document.write( "<font color=blue>B(0)+B(1)+B(2)+B(3)+B(4)+B(5)+B(6)+B(7)</font> = 1-(<font color=red>B(8)+B(9)</font>)\r<BR>\n" );
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document.write( "Rephrased another way: The events \"<font color=blue>at most 7</font>\" and \"<font color=red>more than 7</font>\" are complementary. <BR>\n" );
document.write( "One or the other must happen. <BR>\n" );
document.write( "This is why the terms marked in blue and the terms in red add to 1.\r<BR>\n" );
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document.write( "We could rewrite those previous equations by saying<BR>\n" );
document.write( "<font color=blue>P(x ≤ 7)</font> + <font color=red>P(x > 7)</font> = 1<BR>\n" );
document.write( "<font color=blue>P(x ≤ 7)</font> = 1 - <font color=red>P(x > 7)</font>\r<BR>\n" );
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document.write( "Let's compute B(8)<BR>\n" );
document.write( "B(x) = (9 C x)*(0.75^x)*(0.25)^(9-x)<BR>\n" );
document.write( "B(8) = (9 C 8)*(0.75^8)*(0.25)^(9-8)<BR>\n" );
document.write( "B(8) = 9*(0.75^8)*(0.25)^(1)<BR>\n" );
document.write( "B(8) = 0.2252540588379<BR>\n" );
document.write( "B(8) = 0.22525\r<BR>\n" );
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document.write( "Let's compute B(9)<BR>\n" );
document.write( "B(x) = (9 C x)*(0.75^x)*(0.25)^(9-x)<BR>\n" );
document.write( "B(9) = (9 C 9)*(0.75^9)*(0.25)^(9-9)<BR>\n" );
document.write( "B(9) = 1*(0.75^9)*(0.25)^(0)<BR>\n" );
document.write( "B(9) = 0.0750846862793<BR>\n" );
document.write( "B(9) = 0.07508 \r<BR>\n" );
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document.write( "Sum the results<BR>\n" );
document.write( "B(8)+B(9) = 0.22525 + 0.07508 = 0.30033\r<BR>\n" );
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document.write( "Subtract from 1<BR>\n" );
document.write( "1-0.30033 = 0.69967\r<BR>\n" );
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document.write( "Therefore,<BR>\n" );
document.write( "B(0)+B(1)+B(2)+B(3)+B(4)+B(5)+B(6)+B(7) = 0.69967 approximately.<BR>\n" );
document.write( "That then rounds to <font color=red>the final answer 0.70</font> when rounding to two decimal places.\r<BR>\n" );
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document.write( "There's roughly a 70% chance that at most 7 seeds (i.e. 7 or fewer) will germinate.\r<BR>\n" );
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document.write( "Here are a few binomial distribution calculators.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.gigacalculator.com/calculators/binomial-probability-calculator.php\">https://www.gigacalculator.com/calculators/binomial-probability-calculator.php</a><BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.omnicalculator.com/statistics/binomial-distribution\">https://www.omnicalculator.com/statistics/binomial-distribution</a><BR>\n" );
document.write( "You could also use a spreadsheet or a TI83/TI84 calculator.\r<BR>\n" );
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document.write( "Here is an article talking about binomial probabilities using a TI84<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.statology.org/binomial-probabilities-ti-84-calculator/\">https://www.statology.org/binomial-probabilities-ti-84-calculator/</a><BR>\n" );
document.write( "For this particular problem, we can input <font color=red>binomcdf(9, 0.75, 7)</font> into the TI84. This will add up the values from B(0) to B(7).<BR>\n" );
document.write( "Be sure to use the CDF and not the PDF.<BR>\n" );
document.write( "The PDF is one specific value, while the CDF adds up multiple values below a specific x value.\r<BR>\n" );
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document.write( "Another question involving the binomial probability distribution<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1201351.html\">https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1201351.html</a><BR>\n" );
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