document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1157749>Question 1157749</A>: <I><SPAN STYLE=\"word-break: break-all;\"> suppose that the value of a stock varies each day from $12 to $27 with a uniform distribution . Find the probability that the stock is less than $22 given that the stock is greater than $16. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #780634 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=780634\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer in fraction form = <font color=red>6/11</font><BR>\n" );
document.write( "Approximate answer in decimal form = <font color=red>0.54545</font><BR>\n" );
document.write( "Approximate answer in percent form = <font color=red>54.545%</font>\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "The uniform distribution is graphed as a rectangle. In this problem, the base of the rectangle goes from 12 to 27, which is a distance of 27-12 = 15 units. The height must be 1/15 to ensure that the area of the rectangle is 1.\r<BR>\n" );
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document.write( "area = base*height<BR>\n" );
document.write( "area = 15*(1/15)<BR>\n" );
document.write( "area = 1\r<BR>\n" );
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document.write( "The unit 1 refers to 100% probability\r<BR>\n" );
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document.write( "We are told that \"given that the stock is greater than $16.\" which means we only focus on the portion of the rectangle from 16 to 27. The base is now 27-16 = 11 while the height remains the same\r<BR>\n" );
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document.write( "area = base*height<BR>\n" );
document.write( "area = 11*(1/15)<BR>\n" );
document.write( "area = 11/15\r<BR>\n" );
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document.write( "If we were to throw a random dart at the entire distribution, then the probability of landing on a stock price larger than $16 is 11/15\r<BR>\n" );
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document.write( "Let A = 11/15, which we'll use later.\r<BR>\n" );
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document.write( "Now we ask to find \"the probability that the stock is less than $22\". Keep in mind that we know the stock is greater than $16, so we consider the range from 16 to 22 which has a base of 22-16 = 6 and a height of 1/15, so\r<BR>\n" );
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document.write( "area = base*height<BR>\n" );
document.write( "area = 6*(1/15)<BR>\n" );
document.write( "area = 6/15\r<BR>\n" );
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document.write( "Let B = 6/15\r<BR>\n" );
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document.write( "Now divide B over A to get the conditional probability we're after\r<BR>\n" );
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document.write( "B/A = (6/15) divided by (11/15)<BR>\n" );
document.write( "B/A = (6/15) * (15/11)<BR>\n" );
document.write( "B/A = (6*15)/(15*11)<BR>\n" );
document.write( "B/A = 6/11<BR>\n" );
document.write( "B/A = 0.54545 approximately\r<BR>\n" );
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document.write( "Here's a visual. We know the stock is above $16, which is represented by this shaded red portion below<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/Bg4tSTx.png\"><BR>\n" );
document.write( "This red rectangle has base 11 and height 1/15, so its area is 11/15\r<BR>\n" );
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document.write( "If we want to find the probability of throwing a random dart in the purple region, then we would divide the purple rectangle area over the red rectangle area (from the above figure)<BR>\n" );
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document.write( "This purple rectangle has base 6 and height 1/15, so its area is 6/15\r<BR>\n" );
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document.write( "In other words,<BR>\n" );
document.write( "answer = (area of purple rectangle)/(area of red rectangle)<BR>\n" );
document.write( "answer = (area of rectangle BCFG)/(area of BDEG)<BR>\n" );
document.write( "answer = (6/15)/(11/15)<BR>\n" );
document.write( "answer = 6/11<BR>\n" );
document.write( "answer = 0.54545 approximately\r<BR>\n" );
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document.write( "Another approach:\r<BR>\n" );
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document.write( "Note how the rectangle ADEH is divided into 15*4 = 60 smaller rectangles because of the dashed grid lines<BR>\n" );
document.write( "6*4 = 24 smaller rectangles are painted purple<BR>\n" );
document.write( "11*4 = 44 smaller rectangles are painted red<BR>\n" );
document.write( "24/44 = 6/11 represents the probability of throwing a random dart in the purple region given we know the dart landed in the red region. \r<BR>\n" );
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document.write( "Yet another approach:\r<BR>\n" );
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document.write( "Divide the length of BC over BD<BR>\n" );
document.write( "answer = (length of BC)/(length of BD)<BR>\n" );
document.write( "answer = 6/11\r<BR>\n" );
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document.write( "This is the fastest method, but it does not use the rectangular uniform distribution. So it will depend on what your teacher wants you to show.<BR>\n" );
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