SOLUTION: Me again. Hi - I know this is probably really simple, are I do not know if I am on the right track for this question (probability is just not making any sense to me at all).

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Question 880388: Me again.
Hi - I know this is probably really simple, are I do not know if I am on the right track for this question (probability is just not making any sense to me at all).
It is known that 25% of all staff in a company make use of the free influenza vaccination programme. Answer the following as: The probability is equal to: (Round your probability to 4 decimal places:
Q If 20 people are randomly selected, find the probability that exactly 12 people don't use the vaccination programme.

Please help!
Thanks in advance!
Found 2 solutions by ewatrrr, KMST:
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!
p(make use of vaccine) = .25 n = 20
P(x=12) = 20C12(.25)^12(.75)^8 0r binompdf(20, .25, 12)

Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
I think we are all reading it wrong.
THE ANSWER:
Maybe you just want the answer. In that case, I calculated the approximate probability that 12 of those 20 people got vaccinated as
%2220+C12%220.25%5E12%2A0.75%5E8=0.0008 , but I think you may really have meant what is the probability that 12 of the 20 were not vaccinated.
Then, it would be
%2220+C12%220.25%5E8%2A0.75%5E12=0.0609

THE CALCULATION:
Maybe you want to know how to calculate it.
If you have a computer with the right software, or a calculator with enough functions, you can calculate combinations of 20 taking 12 at a time as 20C12=%28matrix%282%2C1%2C20%2C12%29%29=125970 .
With a simpler calculator, or (gasp!) with pencil and paper,
you would calculate it as
20C12=20C8= .
With most calculators or using your computer you can calculate 0.25%5E8=1.5259%2A10%5E%28-5%29 (approximately) and
0.75%5E12=0.031676 (approximately).
Then you calculate your approximate probability as
125970%2A5.96%2A10%5E%28-8%29%2A0.100=0.06088685426148 , which rounds to 0.0609 .
With pencil and paper it gets a little more cumbersome.

THE REASON WHY YOU APPLY "THE BINOMIAL DISTRIBUTION":
Maybe you want to understand why.
You can describe all the possible outcomes of randomly picking 1 of the people in the staff of that company by the binomial
0.25v%2B0.75n , meaning that
there is a 0.75 probability that he/she got the vaccine and
there is a 0.75 probability that he/she did not get the vaccine.
If you pick 2 people, you can describe all the possible outcomes as the square of that binomial:
.
That means there is a 0.0625 probability that both goth the vaccine;
there is a 0.375 probability that one or the other (got the vaccine(but not both),
and there is a 0.5625 probability that both are unvaccinated.
If you pick 20 people, the whole spectrum of probabilities is described by the 20th power of that binomial:

In that long polynomial, the coefficient of the term with v%5E12 represents the probability that exactly 8 of those 20 people were vaccinated, and the other 12 did not use he vaccination programme.