SOLUTION: hey i would highly appreciate it if you could help me with this complex probability question: A choir consists of 20 girls and 15 boys. A committee of 4 people is selected from

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Question 157871: hey i would highly appreciate it if you could help me with this complex probability question:
A choir consists of 20 girls and 15 boys. A committee of 4 people is selected from the choir.
a) What is the probability that there are only boys on the committee?
b) What is the probability that there is at least 1 boy on the committee?
c) What is the probability that there are exactly 3 girls on the committee?
Found 2 solutions by gonzo, scott8148:
Answer by gonzo(654) (Show Source):
You can put this solution on YOUR website!
question should probably state at random, i.e. all persons in the choir have equal chance of being picked.
probability of picking all boys would be (15/35)*(14/34)*(13/33)*(12/32)
since first pick there are 15 boys out of 35 total.
second pick there are 14 boys out of 34 total.
third pick there are 13 boys out of 33 total.
fourth pick there are 12 boys out of 32 total.
multiplying out probability becomes (15*14*13*12)/35*34*33*32) = .026069519 or approximately 2.6%
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.
probability of all girls is similar to probability of all boys except we start with 20 girls rather than 15 boys.
p = (20*19*18*17)/(35*34*33*32) = .092532468 or approximately 9.3%.
this is the probability of all girls.
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.
probability of exactly 3 girls would be probability of 1 boy plus the probability of 3 girls.
boy could be picked first or girl could be picked first.
p(bggg)first follows:
(15*20*19*18)/(35*34*33*32)
p (gbgg)next follows:
(20*15*19*18)/(35*34*33*32)
p (ggbg)next follows:
(20*19*15*18)/(35*34*33*32)
p(gggb) next follows:
(20*19*18*15)/35*34*33*32)
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.
each one of those probabilities is the same so the sum is 1 * any one of the probabilities.
probability of (gggb) = .081646295 = approximately 8.2%
4 * .081646295 = .32658518 = approximately 32.7%
4 possibilities were chosen by looking at the possible combinations that 3 girls in a committee of 4 could be chosen.
the formula was C(3,4) which equaled (4*3*2)/(1*2*3) which equaled 4.
testing out provided the fact that the formula was correct because the possible combinations looked like
gggb
ggbg
gbgg
bggg
with only 1 boy, 4 was the max combinations that could be generated.
each combination had a unique probability, the sum of which would equal the probability that exactly 3 girls would be chosen.

Answer by scott8148(6628) (Show Source):
You can put this solution on YOUR website!
how many groups of 4 out of 35 people __ 35C4 __ 52360

a) how many groups of 4 out of 15 __ 15C4 __ 1365
__ probability of all boys __ 1365/52360=.026 (approx)

b) how many groups of 4 out of 20 people __ 20C4 __ 4845
__ probability of all girls __ 4845/52360 __ .093 (approx)
__ probability of at least 1 boy __ 1 minus probability of no boys (all girls) __ 1-.093=.907

c) how many groups of 3 out of 20 people __ 20C3 __ 1140
__ each group of 3 girls with any of 15 boys __ 1140*15=17100
__ probability of exactly 3 girls __ 17100/52360=.327 (approx)