document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197430>Question 1197430</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Forty-three percent of marriages end in divorce. You randomly select 15 married couples. Find the expected number of marriages that will end in divorce.\r<BR>\n" );
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document.write( "Suppose an exam consisted of 10 multiple choice problems, each with five possible responses (A-E), only 1 of which is correct.  If a student randomly guesses the answers to each question then what is the probability that a student guesses the correct answer to exactly 7 questions?  Additionally, what is the probability that a student passes the exam with a score of 70% or higher? (round to 5 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 #830698 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</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=830698\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "In the future, please post <u>one</u> problem at a time. \r<BR>\n" );
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document.write( "Question 1) <font color=blue>Forty-three percent of marriages end in divorce. You randomly select 15 married couples. Find the expected number of marriages that will end in divorce.</font>\r<BR>\n" );
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document.write( "This is a binomial distribution problem since we have the following conditions<ul><li>There are two outcomes: The marriage ends in divorce, or it doesn't</li><li>Each couple is independent of one another</li><li>The probability of divorce is the same</li></ul>In this case we have,<BR>\n" );
document.write( "n = 15 married couples = sample size<BR>\n" );
document.write( "p = 0.43 = probability a marriage ends in divorce\r<BR>\n" );
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document.write( "n*p = expected number of marriages ending in divorce<BR>\n" );
document.write( "n*p = 15*0.43<BR>\n" );
document.write( "n*p = 6.45<BR>\n" );
document.write( "Do not round this value since averages aren't always expected to be whole numbers. <BR>\n" );
document.write( "However, if your teacher instructs you to round to the nearest whole number, then be sure to do so.\r<BR>\n" );
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document.write( "Answer: <font color=red>6.45</font>\r<BR>\n" );
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document.write( "Question 2) <font color=blue>Suppose an exam consisted of 10 multiple choice problems, each with five possible responses (A-E), only 1 of which is correct.  If a student randomly guesses the answers to each question then what is the probability that a student guesses the correct answer to exactly 7 questions?  Additionally, what is the probability that a student passes the exam with a score of 70% or higher? (round to 5 decimal places)</font>\r<BR>\n" );
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document.write( "I'll break this up into part (a) and part (b)\r<BR>\n" );
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document.write( "Part (a) will handle the portion <font color=blue>what is the probability that a student guesses the correct answer to exactly 7 questions? </font> and part (b) will handle <font color=blue>what is the probability that a student passes the exam with a score of 70% or higher?</font>\r<BR>\n" );
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document.write( "Part (a)\r<BR>\n" );
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document.write( "There are 5 possible responses (A through E), but only one of which is correct.<BR>\n" );
document.write( "If a student guesses at complete random, then each answer choice is likely to be selected. <BR>\n" );
document.write( "The probability of success here is p = 1/5 = 0.2\r<BR>\n" );
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document.write( "There are n = 10 questions total to make up the sample size.\r<BR>\n" );
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document.write( "We wish to determine the probability of the student getting exactly x = 7 questions correct.\r<BR>\n" );
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document.write( "Use the binomial probability formula<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+expr%28%2810%21%29%2F%28x%21%2A%2810-x%29%21%29%29%2A%280.2%29%5Ex%2A%281-0.2%29%5E%2810-x%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%287%29+=+expr%28%2810%21%29%2F%287%21%2A%2810-7%29%21%29%29%2A%280.2%29%5E7%2A%281-0.2%29%5E%2810-7%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%287%29+=+120%2A%280.2%29%5E7%2A%281-0.2%29%5E%2810-7%29 ALIGN=MIDDLE  ALT=\"f%287%29+=+120%2A%280.2%29%5E7%2A%281-0.2%29%5E%2810-7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%287%29+=+0.000786432 ALIGN=MIDDLE  ALT=\"f%287%29+=+0.000786432\"   DISABLED_style= \"max-width:90%; height:auto;\" > This value is exact\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%287%29+=+0.00079 ALIGN=MIDDLE  ALT=\"f%287%29+=+0.00079\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rounding to five decimal places\r<BR>\n" );
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document.write( "Answer: <font color=red>0.00079</font> (approximate)\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "We need to use the binomial probability formula to find these values of f(x)<BR>\n" );
document.write( "f(7), f(8), f(9), f(10)<BR>\n" );
document.write( "Then we add them all up to find the probability x = 7 or larger.<BR>\n" );
document.write( "This in turn will give scores of 7/10 = 0.70 = 70% or larger.\r<BR>\n" );
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document.write( "I won't show the steps for each, since you can follow the template in part (a)<BR>\n" );
document.write( "Here are the results of each to help check your work.<BR>\n" );
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document.write( "<tr><TD >x</TD><TD >f(x)</TD></tr><tr><TD >7</TD><TD >0.000786432</TD></tr><tr><TD >8</TD><TD >0.000073728</TD></tr><tr><TD >9</TD><TD >0.000004096</TD></tr><tr><TD >10</TD><TD >0.000000102</TD></tr>\n" );
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document.write( "Then add up the results in the f(x) column<BR>\n" );
document.write( "0.000786432+0.000073728+0.000004096+0.000000102 = 0.000864358\r<BR>\n" );
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document.write( "This then rounds to 0.00086 when rounding to five decimal places.\r<BR>\n" );
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document.write( "A quick shortcut is to use a binomial calculator such as this one<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://stattrek.com/online-calculator/binomial\">https://stattrek.com/online-calculator/binomial</a><BR>\n" );
document.write( "or this<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( "If you have a TI83 or TI84, you can use the <font color=blue>binomcdf</font> function. This is found by pressing the button labeled \"2nd\" and then hitting the VARS key. Scroll down a bit until you find the function. <BR>\n" );
document.write( "Refer to this page for more information<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>\r<BR>\n" );
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document.write( "Another alternative is to use a spreadsheet. This is probably the best option considering spreadsheets are used all the time in many real world situations.<BR>\n" );
document.write( "The command to type in would be <font color=blue>=1-BINOM.DIST(6,10,0.2,1)</font> or <font color=blue>=1-BINOMDIST(6,10,0.2,1)</font> is equivalent.<BR>\n" );
document.write( "Don't forget about the equal sign up front.<BR>\n" );
document.write( "The BINOM.DIST(6,10,0.2,1) portion adds up the probabilities f(0), f(1), ... all the way up to f(6). Subtracting the result from 1 will get us the sum of f(7) to f(10).<BR>\n" );
document.write( "Refer to your spreadsheet's documentation for more info on how the BINOMDIST command works.\r<BR>\n" );
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document.write( "Answer: <font color=red>0.00086</font> (approximate)<BR>\n" );
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