Background:
Tyrion, Cersei, and n other guests arrive at a party at i.i.d. times drawn from a continuous distribution with support [0; 1], and stay until the end (time 0 is the party’s start time and time 1 is the end time). The party will be boring at times when neither Tyrion nor Cersei is there, fun when exactly one of them is there, and awkward when both Tyrion and Cersei are there
(b) Jaime and Robert are two of the other guests. By computing both sides in the definition of independence, determine whether the event Jaime arrives at a fun time" is independent of the event Robert arrives at a fun time".
(c) Give a clear intuitive explanation of whether the two events from (b) are independent, and whether they are conditionally independent given the arrival times of everyone else, i.e., everyone except Jaime and Robert.
Solution to (b): Let R and J be that Jaime and Robert arrive in fun time. We are then evaluating if the following equality holds: $P(J,R)= P(J)P(R)$. However, by symmetry of continuous random variables all orderings are equally likely so we have the following probabilities that shows that J and R are not independent.
$$ \begin{align} P(J,R)&= (2\cdot2)/4!=1/6 \\ P(J)P(R) &= 1/3\cdot 1/3 \end{align} $$
No problems here, but the following solution raises an issue.
Solution to (c) (and to my issue...): The solution given is: "Suppose that we know that Jame has come in fun time. That gives less time for Robert to arrive at the party in fun time so the probability for Robert shrinks. On the other hand, if we know the times of arrivals of every other guest, intuitively we can conclude that these events are independent since the arrival of Robert is independent of the time of Jamie's arrival "
However, I was trying to verify that J and R are conditionally independent, but I am arriving still at a dependence. To exemplify, let us take the case where Tyrion, T and Cersei, C arrive in two-distance away from each other (leaving room for two guests falling in between their arrivals and thus creating a "fun party"). Let us make room for five guests in total. Then, fun time is the positions are in between T and C, like so
T _ _ C _
_T _ _ C
and the reversed
C _ _ T _
_C _ _ T
Based on these permutations, we calculate the following conditional probabilities to evaluate if they are equal (and $K_2$ is the event where the positions of T and C are two steps away ) : $P(J,R|K_2)=P(J|K_2)P(R|K_2)$
As we see below, in this particular scenario these events are not conditionally independent.
$$
\begin{align}
P(J,R|K_2)&= \frac{P(J,R,K_2)}{P(K_2)} = 1/3\\
P(J|K_2)P(R|K_2) &= \frac{P(J,K_2)}{P(K_2)} \cdot \frac{P(R,K_2)}{P(K_2)}= 2/3\cdot 2/3
\end{align}
$$
I am calculating these numbers using symmetry of iid continuous random variables, which basically says "all permutations are equally likely". So
For $P(J,R|K)=\frac{2\cdot2\cdot2\cdot1}{5\cdot4\cdot3\cdot2}\cdot 1/\frac{2\cdot2\cdot3\cdot2}{5\cdot4\cdot3\cdot2}=1/3$
What is not adding up to make conditional independence hold?