Confusion on events being dependent or independent (in an example). Background: I have learnt that if events are disjoint they must be dependent. 

I am stuck with this example answer that I have found, here it is: 

In every round of a game, $n$ players throw a legit coin (half probability for head and half for tails.), independently. if one of the players gets a different result from all the other players, he gets out of the game, else they start a new round again, the game keeps going until there's 2 players left which are the winners. 
Q: For a general $n$, what are the expected value, variance and sd of number of rounds that will take place. 

Now, I was looking at the answer and here's what they did : 
We define $Y_k$ as number of rounds that took place while there was $k$ players, until someone gets out. 
And here's my confusion: 
They said that any set of games is independent with the other sets of games (since their intersection is empty), specially for $Y_3, Y_4,...,Y_{n}$. 
My Question: Since the intersection is empty, the sets are disjoint, and that means they're dependent, so why are they using that to say the sets are independent, what am I missing?
 A: The two sets $\{Y_1,Y_2\}$ and $\{Y_2,Y_3\}$ are not disjoint.
The sets $\{Y_1\}$ and $\{Y_2\}$ are disjoint. And the two random variables $Y_1,Y_2$ are independent.
We have four possible outcomes:
$$
\begin{array}{lcl}
Y_1 = 1 & \text{and } & Y_2 = 1 \\[6pt]
Y_1 = 1 & \text{and } & Y_2 = 0 \\[6pt]
Y_1 = 0 & \text{and } & Y_2 = 1 \\[6pt]
Y_1 = 0 & \text{and } & Y_2 = 0 
\end{array}
$$
The two events $\big[ Y_1=1\big]$ and $\big[ Y_2=1 \big]$ are not disjoint since they can both happen, and do both happen when the outcome is $\big[ Y_1=1\text{ and } Y_2=1\big].$ And they are independent.
An event is a set of outomes. Above, four outcomes are listed. As a set of outcomes, the event $\big[ Y_1=1\big]$ is the set whose members are the first two of the four listed outcomes, and the event $\big[Y_2=1\big]$ is the set whose members are the first and third of the listed outcomes. Those two events are independent of each other. They are not disjoint since they have one member in common, namely the first of the four listed outcomes.
Contrast this with the fact that the two random variables $Y_1+Y_2$ and $Y_2+Y_3$ are not independent precisely because the two sets $\{Y_1,Y_2\}$ and $\{Y_2,Y_3\}$ are not disjoint.
Events, i.e. sets of outcomes, cannot be independent if they are disjoint (except when the probability of at least one of them is $0$). Sets of random variables are another matter.
