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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 occurence of
q^{k-r}p^r
because the trials are independant 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?
We define a function P on \Omega by saying say that an elementary event that is a k-tuple has a probability of occurence of
q^{k-r}p^r
because the trials are independant 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?
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