Assume we have a car and a bike. We know the initial, but not future positions of the bike and the current and future positions of the car. Moreover, the bike can only cover a finite distance in one step. For each of the $N$ steps, we can have the events $S_i$ (no collision in step $i$) and $F_i$ (collision in step $i$). Further, say that as soon as $F_i$ occurs for the first time, we stop the experiment. Lastly, let $P(S_i)$ and $P(F_i)$ denote the respective probabilities.
I want to check if $$ P(S_1,S_2,...,S_N) = \prod_{i=1}^{N}P(S_i). $$
For this to hold, I e.g. need that $$ P(S_2|S_1) = P(S_2). $$
I am somewhat struggling with it: The only thing I know from $S_1$ occuring in step one is that car and bike are at least not touching, that is not colliding. In other words, they have to be a certain distance apart. But I feel like calculating $P(S_2)$ already implies that $S_1$ occured. Does it make sense to say that only $P(S_2|S_1)$ exists, but not $P(S_2)$ itself?