Transformation of probability of occurrence to prob of first occurrence I have a series of probabilities that event $E$ occurs at iteration $i$ with $i=1,2,....$. These are denoted $p_1,p_2,....$. These probabilities are that $E$ occurs on iteration $i$, but do not consider whether $E$ occurred before $i$. 
I want to transform these $p$s into a new series such that $\hat{p}^i$ = probability of first occurrence of $E$ on $i$. My thought is that this is:
$$\hat{p}^i = (1-p_1)(1-p_2)....(1-p_{i-1})p_i$$
Assuming $p_i$ is independent of $p_j$ for $i>j$. 
But how do I transform if they are not independent?
In my problem, the probability $E$ occurs over time approaches a stationary distribution and is increasing up to that point. E.g, after some iteration $T$ $p_{t} = X \forall t>T$. "An event waiting to happen"
 A: They could still be independent.  I can come up with an example, although very contrived.
Suppose that p-primes are 0.1, 0.2, 0.3, 0.4 (have to sum to 1)
We can work backwards and come up with ps assuming independence to match using:
$$p'(i)=\prod_{j=1}^{i-1}(1-p(j))\cdot p(i)$$
$$p(i)=\frac{p(i)}{\prod_{j=1}^{i-1}(1-p(j))}$$
Using Matlab ps are 0.1000, 0.2222, 0.4286, 1.0000
A: You would normally talk about independent events rather than probabilities.
To be more precise, let $E_i$ be a binary indicator such that
$$E_i=\begin{cases}
1 & \text{an event occurs at iteration }i \\
0 & \text{otherwise}
\end{cases}$$
Then the marginal probability of $E_i=1$, with respect to an event at some other iteration $j$, will be
\begin{align}\Pr\big[E_i=1\big] &= \Pr\big[E_i=1\mid E_j=1\big]\times\Pr\big[E_j=1\big] \\ &+\Pr\big[E_i=1\mid E_j=0\big]\times\Pr\big[E_j=0\big]\end{align}
If the event $E_i$ is independent of the event $E_j$, then you will have
$$\Pr\big[E_i=1\mid E_j=1\big] = \Pr\big[E_i=1\mid E_j=0\big] = p_i$$
This intuitively would seem to correspond to your statement about "agnostic".
Mathematically, since you are specifying the probabilities $p_i$ without reference to the sequence of values $E_{j<i}$, your model is assuming independent events. (Otherwise you would need to marginalize over possible "prefix sequences" to determine the $p_i$.)
