It is sort of like the binomial distribution. Remember with the binomial that we are not just interested in say (in your case) QQQQQP, where Q is a failure, and P is a success, because I could have PQQQQQ, and QQPQQQ, where each arrangement of P's and Q's is the probability of getting one six in six throws. For a binomial, we are interested in ALL possible ways of getting one six in six throws. What the geometric says, is we are interested in that sequence where I have thrown the die 6 times, and I want the probability that the next throw is a success: This would be P(Five failures) and P(Success on Next try); since these two events are independent - we just multiply the two probabilities.

More generally we define a geometric random variable as:

where p is the success probability, and X is the number of throws it takes you to get a success. The probability mass function for the random variable X is then:

. With a little digging you can see that this is a "binomial distribution" with "n choose k" of zero (where "n" is equal to X-1 from the geometric distribution):

, multiplied by "p".

Hopefully this makes some sense and helps to tie all of this together.