Consider a sequence of n independent multinomial trials; in each trial there are three possible outcomes, type 1, type 2, and type 3 with respective probabilities p1, p2, and p3, where p1 + p2 + p3 = 1. Let N$_i$ be the first trial for which an outcome of type i occurs, for i = 1, 2, 3 (e.g., N$_i$ = 1 means the first trial had outcome of type i and N$_i$ = 2 means the first trial did not have outcome of type i but the second trial did).
Question #1: For 1 < j < k, compute P(N$_1$ = 1, N$_2$ = j, N$_3$ = k) in terms of p1, p2, p3. Are N$_1$, N$_2$, N$_3$ mutually independent?
What I tried, was to find the sample space for the event of N$_1$ = 1, N$_2$ = j, N$_3$ = k. (which is quite large) and put it over the total amount of events.
I know we must show whether or not they are mutually independent by: P(N$_1$ = 1, N$_2$ = j, N$_3$ = k) = P(N$_1$)P(N$_2$)P(N$_3$)
My thinking is that they are NOT mutually independent, since if N$_3$ > N$_2$, then we cannot have: N$_2$=j and N$_3$=k, for j < k. Therefore, the event of picking N$_3$ > N$_2$, depends on the trial that N$_2$ is first picked.
Is my thinking wrong?
Indeed they are not independent.
Here is a counterexample:
$$\underbrace{P(N_1=1 \ \ \& \ \ N_2=1)}_{= \ 0 \ \ \text{(non compatible events)}} \ \ \ \ \neq \ \ \ \ \underbrace{P(N_1=1)}_{\neq 0}.\underbrace{P(N_2=1)}_{\neq 0}$$
Remark : Notation "$N_1=1"$ for example is for the non elementary event gathering all elementary events beginning by "$N_1=1"$.