Prove that event B with probability 1 sooner or later happens We have the sequence of independent trials. In any event it may be $A$ or $B$. The probability of an event relating to a particular test is for all the trials the same. Event $B$ happens with strictly positive probability.
Prove that event $B$ with probability $1$ sooner or later happens.
I started like this:
$$P(B) = 1$$
$$P(B)=\sum_{i=1}^{\infty} p^i =p+p^2+p^3+... =1.$$
What should be $p$ so that my sum is correct? I think about $p>0$ or $p>1/2$.
Thank you for your help.
 A: The probability of failing $k$ times is $(1-p)^k$ supposing the trials are independent from each other. If you do more trials $(1-p)^k$ tends to $0$ as the number of trials tends $k$ tends to infinity.  Or in other words the probability of succeeding tends to $1$ as $k$ tends to infinity. 
EDIT: The caveat that @Deepak Gupta added is indeed important, in probability theory we say that something happens with a certain probability. To see why this is important, flip a coin an infinite number of times an write down $0$ for heads and $1$ for tails. It is indeed possible to have an infinite sequence of zeros. The probability that you will have written down this sequence is $0$. So, counter-intuitive  as it may be, the fact that an event has probability $0$ does not mean that it "cannot happen".
EDIT: The reason why your own answer is incorrect is the following: 
It does not make sense you are just adding the probabilities of it happening $k$ times in a row. Furthermore, the answers I and @Deepak Gupta gave showed that $p$ need not be some specific value, it only needs to be strictly larger than $0$.
A: I am not sure I understand the wording of your question and I particularly think that the phrase "with probability 1" is implied by the remaining sentence "sooner or later happens" and hence is redundant.
Let $C$ represent the event that $B$ never happens regardless of the number of independent trials.
Let $P(B)=p$ and $ 0<p \le 1$. Then $P(B')=1-p $ and $ 0 \le 1-p < 1$.
Since the trials are independent events, the probability $P(C)$ that $B$ never occurs is - $$ P(C)=(1-p)^\infty = 0 \,\,\,\,\, \forall \,\,\,\,\,0 \le 1-p<1$$
Now since the probability that $B$ never occurs is zero, it implies that $B$ must occur atleast once sooner or later.
