document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=848062>Question 848062</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Based on past experience, it is known that 10% of newly launched businesses experience cash flow problems. An MBA student selects a random sample of 5 newly launched businesses. Assuming that the distribution of newly launched businesses that experience cash flow problems is independent, \r<BR>\n" );
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document.write( "a) What is the probability that at most 3 of the 5 businesses experience cash flow problems?  \r<BR>\n" );
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document.write( "b) What is the probability that between 3 and 5 businesses experience cash flow problems?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #510849 by </B><A HREF=/tutors/aboutme.mpl?userid=swincher4391><B>swincher4391(1107)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=swincher4391><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/cgi-bin/show-face.mpl?user=swincher4391><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=510849\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let X be a binomial(p=.10, n = 5)\r<BR>\n" );
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document.write( "P[X<=3]:\r<BR>\n" );
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document.write( "Two options. I suggest using the one that involves less calculations.\r<BR>\n" );
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document.write( "P[X<=3]: P[X=0] + P[X=1] + P[X=2] + P[X=3]<BR>\n" );
document.write( "or<BR>\n" );
document.write( "Using the complement of P[X<=3]: 1-P[X>=4] = 1-(P[X=4]+P[X=5])\r<BR>\n" );
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document.write( "I will use the 2nd.\r<BR>\n" );
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document.write( "P[X=4] = (5 choose 4) * (.10)^4 * (.90)^1 = 0.00045<BR>\n" );
document.write( "P[X=5] = (5 choose 5) * (.10)^5 * (.90)^0 = 0.00001\r<BR>\n" );
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document.write( "1 - (0.00045 + 0.00001) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28.9995%29 ALIGN=MIDDLE  ALT=\"highlight%28.9995%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Question 2:\r<BR>\n" );
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document.write( "P[3<=X<=5] =  P[X=3] + P[X=4] + P[X=5]\r<BR>\n" );
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document.write( "or  1-(P[X=0] + P[X=1] + P[X=2])\r<BR>\n" );
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document.write( "The first offers less calculations, and we've already calculated 2/3 of it in the first one. All that is required is to calculate \r<BR>\n" );
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document.write( "P[X=3] = (5 choose 3) * (.10)^3 *(.90)^2 = .0081\r<BR>\n" );
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document.write( "P[X=3 or 4 or 5] = .0081 + .00045 + .00001 = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28.0086%29 ALIGN=MIDDLE  ALT=\"highlight%28.0086%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Just to make sure, I'll calculate using the complement as well.\r<BR>\n" );
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document.write( "P[X=0] = (5 choose 0)*(.10)^0*(.90)^5 = .5905\r<BR>\n" );
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document.write( "P[X=1] = 5 * .10^1  * .9^4 = .3281\r<BR>\n" );
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document.write( "P[X=2] = 10 * .10^2 * .9^3 = .0729\r<BR>\n" );
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document.write( "1 - .5905 - .3281 - .0729 =  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28.0085%29 ALIGN=MIDDLE  ALT=\"highlight%28.0085%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <-close enough due to some rounding\r<BR>\n" );
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document.write( "  \r<BR>\n" );
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