document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=697630>Question 697630</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The probability that a certain machine turns out a defective item is .05. Find the probabilities that in a run of 75 items, the following results are obtained.\r<BR>\n" );
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document.write( "a. Exactly 5 defective items\r<BR>\n" );
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document.write( "b. No defective items\r<BR>\n" );
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document.write( "c. At least 1 defective item </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #430354 by </B><A HREF=/tutors/aboutme.mpl?userid=Positive_EV><B>Positive_EV(69)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Positive_EV><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=Positive_EV><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=430354\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The distribution of the number of defective items in this run is binomial with n = 75 items and p = .05 of each item being defective. The probability mass function of the binomial distribution is:<br>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X+=+k%29+=+%28nCk%29%2A%28p%5Ek%29%2A%281-p%29%5E%28n-k%29 ALIGN=MIDDLE  ALT=\"P%28X+=+k%29+=+%28nCk%29%2A%28p%5Ek%29%2A%281-p%29%5E%28n-k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where nCk is the number of combinations of k objects chosen from n = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%21%2F%28k%21%2A%28n-k%29%21%29 ALIGN=MIDDLE  ALT=\"n%21%2F%28k%21%2A%28n-k%29%21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
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document.write( "1) In this case, we set k = 5. n = 75 and p = .05, so:<br>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X+=+5%29+=+%2875C5%29%2A%28.05%5E5%29%2A%28.95%29%5E%2870%29 ALIGN=MIDDLE  ALT=\"P%28X+=+5%29+=+%2875C5%29%2A%28.05%5E5%29%2A%28.95%29%5E%2870%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X+=+5%29+=+%2875%21%2F%285%21%2A%2870%29%21%29%29%2A%28.05%5E5%29%2A%28.95%29%5E%2870%29 ALIGN=MIDDLE  ALT=\"P%28X+=+5%29+=+%2875%21%2F%285%21%2A%2870%29%21%29%29%2A%28.05%5E5%29%2A%28.95%29%5E%2870%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28X+=+5%29+=+%2875%21%2F%285%21%2A%2870%29%21%29%29%2A%28.05%5E5%29%2A%28.95%29%5E%2870%29+=+.1488 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
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document.write( "2) In this case, we set k = 0. The binomial probability mass function reduces to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281-p%29%5En ALIGN=MIDDLE  ALT=\"%281-p%29%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" > when k = 0, so P(X = 0) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28.95%29%5E75+=+.0213 ALIGN=MIDDLE  ALT=\"%28.95%29%5E75+=+.0213\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br>\r<BR>\n" );
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document.write( "3) The event that at least one item is defective is the compliment of the event that there are no defective items. The probability of a complimentary event happening is 1 - P(original event), so the probability of at least one defective item is 1 - .0213 = .9787. </SPAN>\n" );
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