I am looking for a solution to this problem:
A dating service finds that 15% of all couples it matches eventually
get married. In the next 50 matches that the service makes, find the
probability that at least 6 couples marry.

There a formula for doing this called the Binomial Probability Formula:
http://www.mathwords.com/b/binomial_probability_formula.htm

The formula looks like:

  P(x) = nCx * p^x * q^(n-x)

  n = number of trials
  x = number of successes among n trials
  p = probability of success in any one trial
  q = probability of failure in any one trial (q = 1 - p)
  nCx = combinations of n items, choose x

This gives you P(6 out of 50 get married) = 50C60 * .15^6 * .85^(50-6)

The first thing to do is to figure out how many different combinations there are.

So you need to know how many different ways there are for 6 couples to
be chosen out of 50, or 50 choose 6.  There happens to be formula for
doing that:

http://mathforum.org/dr.math/faq/faq.comb.perm.html
So...we get 50!/6!(50-6)!, this works out to be 15890700

So P(6 of 50 get married) = 15890700 * .15^6 * .85^(50-6)

P(6 of 50 get married) ~ 14%

Woops...that's 50C6 not 50C60...typo...doesn't effect the answer though...

Most of the previous comment is correct with a minor correction. Since
the answer requires to find the probability of atleast 6 marriages,
the correct answer would be

P(at least 6 marriages) = Sum(P(n)), where n = 6 .. 50
                     OR = 1 - Sum (P(n)), where n = 0 .. 5

doh!  you're right...didn't notice that at least 6 part.

When you work out the math, the final answer should come out to about 78%.

Thanks Guru-Ghantal and Feldersoft

The reason I am asking these is because I am trying to impress a women
who is having probability as a paper in school and she thinks that I
am very math type person.

So be prepared for more questions coming my way, ofcourse, she will
submit it to her professor who will be evaluating it.

But many thanks guys.

Regards