Is P(\Omega) Equal to 1 in Bernoulli Trials?

[quasar987]

Consider a sequence of Bernoulli trials where we play until the rth success is attained. Denote $\Omega$ the fundamental set.

We define a function P on $\Omega$ by saying say that an elementary event that is a k-tuple has a probability of occurrence of

$$q^{k-r}p^r$$

because the trials are independent and the probability of success at each of them is p and the probability of failure is q.

Now I ask wheter or not with these probabilities assigned to each elementary event, we do have $P(\Omega)=1$? Did I miss something and it is implied that $P(\Omega)=1$, or we have to show that

$$\sum_{k=r}^{\infty}\binom{k-1}{r-1}q^{k-r}p^r=1$$

to show that P defined above is indeed a probability on $\Omega$?

 

[marcmtlca]

that looks like a negative binomial distribution... not sure. but you should probably prove it sums to 1.