document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=157871>Question 157871</A>: <I><SPAN STYLE=\"word-break: break-all;\"> hey i would highly appreciate it if you could help me with this complex probability question:\r<BR>\n" );
document.write( "\n" );
document.write( "A choir consists of 20 girls and 15 boys. A committee of 4 people is selected from the choir.\r<BR>\n" );
document.write( "\n" );
document.write( "a) What is the probability that there are only boys on the committee?\r<BR>\n" );
document.write( "\n" );
document.write( "b) What is the probability that there is at least 1 boy on the committee?\r<BR>\n" );
document.write( "\n" );
document.write( "c) What is the probability that there are exactly 3 girls on the committee? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #116341 by </B><A HREF=/tutors/aboutme.mpl?userid=gonzo><B>gonzo(654)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=gonzo><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=116341\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> question should probably state at random, i.e. all persons in the choir have equal chance of being picked.<BR>\n" );
document.write( "probability of picking all boys would be (15/35)*(14/34)*(13/33)*(12/32)<BR>\n" );
document.write( "since first pick there are 15 boys out of 35 total.<BR>\n" );
document.write( "second pick there are 14 boys out of 34 total.<BR>\n" );
document.write( "third pick there are 13 boys out of 33 total.<BR>\n" );
document.write( "fourth pick there are 12 boys out of 32 total.<BR>\n" );
document.write( "multiplying out probability becomes (15*14*13*12)/35*34*33*32) = .026069519 or approximately 2.6%<BR>\n" );
document.write( "probability of at least 1 boy is (1 - the probability of all girls) since it only takes one boy to spoil the all girl show.<BR>\n" );
document.write( "probability of all girls is similar to probability of all boys except we start with 20 girls rather than 15 boys.<BR>\n" );
document.write( "p = (20*19*18*17)/(35*34*33*32) = .092532468 or approximately 9.3%.<BR>\n" );
document.write( "this is the probability of all girls.<BR>\n" );
document.write( "probability of at least 1 boy is 1 - the probability of all girls so the answer is approximately 90.7% that at least 1 boy will be on the committee.<BR>\n" );
document.write( "probability of exactly 3 girls would be probability of 1 boy plus the probability of 3 girls.<BR>\n" );
document.write( "boy could be picked first or girl could be picked first.<BR>\n" );
document.write( "p(bggg)first follows:<BR>\n" );
document.write( "(15*20*19*18)/(35*34*33*32)<BR>\n" );
document.write( "p (gbgg)next follows:<BR>\n" );
document.write( "(20*15*19*18)/(35*34*33*32)<BR>\n" );
document.write( "p (ggbg)next follows:<BR>\n" );
document.write( "(20*19*15*18)/(35*34*33*32)<BR>\n" );
document.write( "p(gggb) next follows:<BR>\n" );
document.write( "(20*19*18*15)/35*34*33*32)<BR>\n" );
document.write( "there are 4 ways in which 1 boy and 3 girls can be chosen so the probability would be the sum of those 4 probabilities.<BR>\n" );
document.write( "each one of those probabilities is the same so the sum is 1 * any one of the probabilities.<BR>\n" );
document.write( "probability of (gggb) = .081646295 = approximately 8.2%<BR>\n" );
document.write( "4 * .081646295 = .32658518 = approximately 32.7%<BR>\n" );
document.write( "4 possibilities were chosen by looking at the possible combinations that 3 girls in a committee of 4 could be chosen.<BR>\n" );
document.write( "the formula was C(3,4) which equaled (4*3*2)/(1*2*3) which equaled 4.<BR>\n" );
document.write( "testing out provided the fact that the formula was correct because the possible combinations looked like<BR>\n" );
document.write( "gggb<BR>\n" );
document.write( "ggbg<BR>\n" );
document.write( "gbgg<BR>\n" );
document.write( "bggg<BR>\n" );
document.write( "with only 1 boy, 4 was the max combinations that could be generated.<BR>\n" );
document.write( "each combination had a unique probability, the sum of which would equal the probability that exactly 3 girls would be chosen.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );