document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1024805>Question 1024805</A>: <I><SPAN STYLE=\"word-break: break-all;\"> KSDLD-Pros, an IT Project consulting firm, is forming an IT Project management team of 5 professionals. In the firm of 50 professionals, 8 are considered to be data analytics specialists. If the professionals are selected at random, what is the probability that the team will include...<BR>\n" );
document.write( "a. no data analytics specialist?<BR>\n" );
document.write( "b. at least one data analytics specialist?  </SPAN>\n" );
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document.write( "Question:<BR>\n" );
document.write( "KSDLD-Pros, an IT Project consulting firm, is forming an IT Project management team of 5 professionals. In the firm of 50 professionals, 8 are considered to be data analytics specialists. If the professionals are selected at random, what is the probability that the team will include...<BR>\n" );
document.write( "a. no data analytics specialist?<BR>\n" );
document.write( "b. at least one data analytics specialist? <BR>\n" );
document.write( " <BR>\n" );
document.write( "Solution:<BR>\n" );
document.write( "The hypergeometric distribution will be used.<BR>\n" );
document.write( "A=number of Analytics specialists in the population (8)<BR>\n" );
document.write( "B=number of non-Analytics specialists in the population (42)<BR>\n" );
document.write( "a=number of Analytics specialists chosen in the sample (x)<BR>\n" );
document.write( "b=number of non-Analytics specialists chosen in the sample (5-x)<BR>\n" );
document.write( "and<BR>\n" );
document.write( "C(n,r)=n!/(r!(n-r)!) is the binomial coefficient of taking r out of n objects<BR>\n" );
document.write( "then <BR>\n" );
document.write( "P(a,a+b)  [choosing a specialists out of (a+b)]<BR>\n" );
document.write( "=C(a,A)*C(b,B)/C(a+b,A+B)  [hypergeometric distribution]\r<BR>\n" );
document.write( "\n" );
document.write( "(a) No analytics specialist (x=0, a=0, b=5)<BR>\n" );
document.write( "P(a,a+b)<BR>\n" );
document.write( "P(0,5)<BR>\n" );
document.write( "=C(8,0)*C(42,5)/C(50,5)<BR>\n" );
document.write( "=1*850668/2118760<BR>\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30381%2F75670 ALIGN=MIDDLE  ALT=\"30381%2F75670\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "=0.4015 (approx.)<BR>\n" );
document.write( "(b)At least one analytics specialist<BR>\n" );
document.write( "Since this case is the complement of case (a), the result is <BR>\n" );
document.write( "P(1&le;x&le;5;5)<BR>\n" );
document.write( "=1-P(0,5)<BR>\n" );
document.write( "=0.5985 (approx.)<BR>\n" );
document.write( " <BR>\n" );
document.write( "On the other hand, if we would have used (incorrectly) the binomial distribution, then p=0.2, n=0,<BR>\n" );
document.write( "(a) P(0,5)=C(0,5)*0.2^0*(0.8^5)=0.328<BR>\n" );
document.write( " <BR>\n" );
document.write( "This problem has a relatively small population (50)from which a relatively large sample (5) is taken.  Use of the binomial distribution will only provide an approximation, since the probability of choosing a analytics specialist (varies from 20% to a minimum of 11%) varies during the experiment.\r<BR>\n" );
document.write( "\n" );
document.write( "So it is important to choose the right distribution for a given application. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );