Interpretation of cdf of geometric distribution Given a geometric random variable $X$ with $p = 0.05$, I want to find (for example) $P(X \gt 10)$. Trivially, this is $1 - P(X\leq10)$, which can be evaluated with the cdf as $1-0.4013$ or $0.5987$. That means the probability that the number of failures before I get my first success is larger than 10 is about $59.87$%. What I don't get is: $0.5987=0.95^{10}$, or exactly 10 failures...! How is it that this gives me the probability of having more than 10 failures? To me, $0.5688=0.95^{11}$ seems like a much more reasonable value: I got 11 failures, which is the bare minimum for having more than 10 failures... any further failures will be included in my $P(X\gt10)$. I know I'm wrong, but I'd appreciate some help in understanding why.
 A: The event $(X > 10)$ is the union of the disjoint events $(X=11), (X=12),(X=13), \cdots$ and so $P(X>10)$ is the sum of the probabilities of these events. The hard way of calculating $P(X>10)$ is to sum the (geometric) series to arrive at $0.95^{10}$. The easier way to get to the same answer is by musing on the fact that the only way that the event $(X>10)$ can occur, that is, the first success to occur on the 11th or 12th or 13th or... is for the first ten trials to have ended in failure, and this has probability $0.95^{10}$ of occurring.
A: Ok, after reading through the Wikipedia article on the Geometric distribution, I believe I understand the problem. There are two ways to interpret what the Geometric distribution means: (1) the number of trials needed to get the first success; or (2) the number of failures needed before the first success. Both have different CDFs: for (1) it's $P(X \leq k)= 1-(1-p)^k$, and for (2) it's $P(X \leq k)= 1-(1-p)^{k+1}$. The problem I was trying to solve explicitly defined itself as "the number of failures before your first success", or (2), but (for some reason) expected me to solve it using the CDF from (1). For completion, by following the CDF from (2), we get $P(X\gt10)=1-P(X \leq 10)=1-(1-(1-0.05)^{10+1})=0.95^{11}=0.5688$, as I initially expected. On the other hand, the CDF from (1) results in $0.95^{10}$, which is what the problem expected.
