Consider a sequence of Bernoulli trials where we play until the rth success is attained. Denote [itex]\Omega[/itex] the fundamental set.(adsbygoogle = window.adsbygoogle || []).push({});

We define a function P on [itex]\Omega[/itex] by saying say that an elementary event that is a k-tuple has a probability of occurence of

[tex]q^{k-r}p^r[/tex]

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 [itex]P(\Omega)=1[/itex]? Did I miss something and it is implied that [itex]P(\Omega)=1[/itex], or we have to show that

[tex]\sum_{k=r}^{\infty}\binom{k-1}{r-1}q^{k-r}p^r=1[/tex]

to show that P defined above is indeed a probability on [itex]\Omega[/itex]?

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# Probability - Bernoulli stuff

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