# What is a sub-collection of events when trying to define mutual independence?

Let $$H_{1}$$, ..., $$H_{n}$$ be n events. $$H_{1}$$, ..., $$H_{n}$$ are jointly independent (or mutually independent) if and only if for any sub-collection of k events ($$k \leq n$$) $$H{_{i1}}$$, ..., $$H_{i_{k}}$$

My definition of mutual independence: $$P(\cap^{k}_{j=1}E_{1j}) = \Pi^{k}_{j=1}P(E_{ij})$$.

I read this and assumed something along the lines of:

S = {HHH, HHT, HTH, THH, THT, TTH, HTT, TTT}

Let $$H_1$$ be equal to the event of getting a heads on the first toss. So $$H_1$$ = {HHH, HHT, HTT, HTH}.

I then thought that an example of a sub-collection of an event would be $$H_{11} = {HHH}$$. Although, I think I am wrong.

But I do not see how my idea of a sub-collection of an event, helps clear up my definition of mutual independence. I think that I may be reading the definition wrong.

edit: $$E_{12} = HHT, E_{13} = HTT, \ldots$$

Let $$H_1, H_2, H_3$$ be the events we need to be independent. Any subcollection of events means that we can take any of these events. To be independent, they should satisfy four conditions: $$k=2,\; i_1=1,\;i_2=2: \quad \mathbb P(H_1\cap H_2)=\mathbb P(H_1)\mathbb P(H_2).$$ $$k=2,\; i_1=1,\;i_2=3: \quad \mathbb P(H_1\cap H_3)=\mathbb P(H_1)\mathbb P(H_3).$$ $$k=2,\; i_1=2,\;i_2=3: \quad \mathbb P(H_2\cap H_3)=\mathbb P(H_2)\mathbb P(H_3).$$ $$k=3,\; i_1=1,\;i_2=2,\;i_3=3: \quad \mathbb P(H_1\cap H_2\cap H_3)=\mathbb P(H_1)\mathbb P(H_2)\mathbb P(H_3).$$
If you take $$H_1=\{HHH, HHT, HTT, HTH\}$$, you should take the other events $$H_2$$, $$H_3$$ and so on to check independence. Say, $$H_2=\{HHH, HHT, THH, THT\}$$ means that the second coin is head. Let also $$H_3=\{HHH, HTH, THH, TTH\}$$ means that the third coin is head.
Take $$k=2$$ and events with numbers $$i_1=1$$ and $$i_2=2$$. They are $$H_1$$ and $$H_2$$. Their intersection is $$H_1\cap H_2 = \{HHH, HHT\}$$. Probability of this event is $$\frac28=\frac14=\frac12\cdot\frac12=\mathbb P(H_1)\cdot \mathbb P(H_2)$$.
Then take the other pari of events $$H_1$$ and $$H_3$$ and so on.