$P(E_{1}\cap E_{2}\cap ... \cap E_{n})\leq \Pi_{i=1}^{n}P(E_{i})$?

Question

I am working on a probability problem in which there are $n$ events $E_{1},E_{2},...,E_{n}$. I'm interested to know the value or an upper bound of $P(E_{1}\cap E_{2}\cap ... \cap E_{n})$. I've shown that, for these events $E_i$, it holds that

$$P(E_{i}\cap E_{j})\leq P(E_{i})P(E_{j}), \quad 1\leq i<j\leq n \tag{$1$}$$

By using $(1)$, can we conclude $P(E_{1}\cap E_{2}\cap ... \cap E_{n})\leq \Pi_{i=1}^{n}P(E_{i})$?

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For example, let $\Omega = \{1,2,3,4,5,6\}, E_{1} = \{1,2,3\}$ and $E_{2}=\{1,4,5,6\}$. Then $E_{1}\cap E_{2} = \{1\}$, and $$P(E_{1}\cap E_{2})=\frac{1}{6}\leq P(E_{1})P(E_{2})=\frac{1}{2}\cdot\frac{2}{3}=\frac{1}{3}$$

Answer 1

What if $(X_1,X_2,X_3)=(0,0,1)$, $(0,1,0)$, $(1,0,0)$ and $(1,1,1)$ each with probability $\frac14$ and $E_i=\{X_i=1\}$?

Answer 2

Even assuming $(1)$ holds (and in general it doesn't) we cannot prove your statement.

I think it is more clear to first show the wrong proof and then point out where it goes wrong. The "false proof" here is the following: $$P(E_1 \cap \dots \cap E_n) \le P(E_1) P(E_2 \cap \dots \cap E_n) \le P(E_1)P(E_2)P(E_3 \cap \dots \cap E_n) \le \Pi_{i=1}^n P(E_i)$$

Unfortunately, this is wrong because we never proved that $P(E_1 \cap \dots \cap E_n) \le P(E_1)P(E_2 \cap \dots \cap E_n)$. We only proved that for the events $E_i$ themselves, not for their intersections. To make the proof work, you would need to prove that for every $i$,

$$P(E_i \cap E_{i+1} \cap \dots \cap E_n) \le P(E_i)P(E_{i+1}\cap \dots \cap E_n)$$

and unfortunately this is not implied by your $(1)$.

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The confusion may stem from the fact that often we prove statements for every events that belongs to an algebra (or $\sigma$-algebra). If this were the case, a relationship like $(1)$ would be enough, because $E_2 \cap \dots \cap E_n$ is another event in the algebra and we could apply the same formula again. So one way to fix it would be to prove that $E_2 \cap \dots \cap E_n$ still belongs to the family of events for which the property $(1)$ holds. For example:

Let $E$ be the set of all the events $E_i$ for which $(1)$ holds. Note that $E$ contains your events $E_1, \dots, E_n$ but potentially many more. Now prove that the set $E$ is closed under (finite) intersection (Note that $E$ being an algebra would imply this). Then you're good, the proof above works, and your statement is proved! :-)