document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1150084>Question 1150084</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A box of 30 flashbulbs contains 10 defective bulbs. A random sample of 2 is selected and tested. Let X be the random variable associated with the number of defective bulbs in the sample. <BR>\n" );
document.write( "​(A) Find the probability distribution of X. <BR>\n" );
document.write( "​(B) Find the expected number of defective bulbs in the sample. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #771445 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=771445\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"times\" color=\"black\" size=\"3\"><BR>\n" );
document.write( "Part A<BR>\n" );
document.write( "X = number of defective bulbs<BR>\n" );
document.write( "X is an integer chosen from the set {0, 1, 2}.<BR>\n" );
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document.write( "If X = 0, then we have no defective bulbs, which means both bulbs are working.<BR>\n" );
document.write( "The probability of choosing 2 working bulbs is (20/30)*(19/29) = 38/87<BR>\n" );
document.write( "20/30 represents the probability of choosing one working bulb (20 working out of 30 total)<BR>\n" );
document.write( "19/29 represents the probability of choosing a second working bulb (each value goes down by 1 becuse we dont replace whatever bulb was selected)\r<BR>\n" );
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document.write( "So if X = 0, then P(X) = 38/87<BR>\n" );
document.write( "We'll have this in the table below<BR>\n" );
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document.write( "When X = 1, we have one working bulb and one defective bulb<BR>\n" );
document.write( "A = P(first bulb works) = 20/30 = 2/3<BR>\n" );
document.write( "B = P(second bulb doesnt work) = 10/29<BR>\n" );
document.write( "C = A*B = (2/3)*(10/29) = 20/87\r<BR>\n" );
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document.write( "D = P(first bulb doesnt work) = 10/30 = 1/3<BR>\n" );
document.write( "E = P(second bulb works) = 20/29<BR>\n" );
document.write( "F = D*E = (1/3)*(20/29) = 20/87<BR>\n" );
document.write( "We get the same result each time so the order doesnt matter (if we get defective first or not)\r<BR>\n" );
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document.write( "C+F = (20/87)+(20/87) = 40/87 is the probability of having one bulb working and the other defective\r<BR>\n" );
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document.write( "If X = 1, then P(X) = 40/87<BR>\n" );
document.write( "We'll have this in the table below<BR>\n" );
document.write( "---------<BR>\n" );
document.write( "Finally, if X = 2 then we have 2 defective bulbs<BR>\n" );
document.write( "P(both defective) = P(first defective)*P(second defective) = (10/30)*(9/29) = 3/29\r<BR>\n" );
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document.write( "So if X = 2, then P(X) = 3/29<BR>\n" );
document.write( "We'll have this in the table below\r<BR>\n" );
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document.write( "---------<BR>\n" );
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document.write( "The probability distribution looks like this<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
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document.write( "<tr><TD >X</TD><TD >P(X)</TD></tr>\r\n" );
document.write( "<tr><TD >0</TD><TD >38/87</TD></tr>\r\n" );
document.write( "<tr><TD >1</TD><TD >40/87</TD></tr>\r\n" );
document.write( "<tr><TD >2</TD><TD >3/29</TD></tr>\r\n" );
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document.write( "</TABLE>\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "The fractions can be replaced with their decimal approximations<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
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document.write( "<tr><TD >X</TD><TD >P(X)</TD></tr>\r\n" );
document.write( "<tr><TD >0</TD><TD >0.43678</TD></tr>\r\n" );
document.write( "<tr><TD >1</TD><TD >0.45977</TD></tr>\r\n" );
document.write( "<tr><TD >2</TD><TD >0.10345</TD></tr>\r\n" );
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document.write( "</TABLE><BR>\n" );
document.write( "38/87 = 0.43678<BR>\n" );
document.write( "40/87 = 0.45977<BR>\n" );
document.write( "3/29 = 0.10345<BR>\n" );
document.write( "===========================================================<BR>\n" );
document.write( "Part B\r<BR>\n" );
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document.write( "Refer to the table created in part A<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
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document.write( "<tr><TD >X</TD><TD >P(X)</TD></tr>\r\n" );
document.write( "<tr><TD >0</TD><TD >38/87</TD></tr>\r\n" );
document.write( "<tr><TD >1</TD><TD >40/87</TD></tr>\r\n" );
document.write( "<tr><TD >2</TD><TD >3/29</TD></tr>\r\n" );
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document.write( "</TABLE>\r<BR>\n" );
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document.write( "Let's add on a new column called X*P(X). This column is the result of multiplying each X and P(X) value together for any given row<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
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document.write( "<tr><TD >X</TD><TD >P(X)</TD><TD >X*P(X)</TD></tr>\r\n" );
document.write( "<tr><TD >0</TD><TD >38/87</TD><TD >0*(38/87) = 0</TD></tr>\r\n" );
document.write( "<tr><TD >1</TD><TD >40/87</TD><TD >1*(40/87) = 40/87</TD></tr>\r\n" );
document.write( "<tr><TD >2</TD><TD >3/29</TD><TD >2*(3/29) = 6/29</TD></tr>\r\n" );
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document.write( "</TABLE>\r<BR>\n" );
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document.write( "Add up the results in that new column<BR>\n" );
document.write( "0+40/87+6/29 = 40/87+18/87 = 58/87 = (29*2)/(29*3) = 2/3\r<BR>\n" );
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document.write( "Expected number of defective bulbs in the sample = 2/3 = 0.667\r<BR>\n" );
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document.write( "Keep in mind that having a fractional amount of defective bulbs does not make sense for a single sample; however, if you repeatedly pulled out 2 bulbs at random (do this say 1000 times), then you should expect on average to have 0.667 defective bulbs. <BR>\n" );
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