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I came to the following problem: Let $A_1, A_2, ...$ be events in a probability space $(\Omega, F, \mathbb{P})$ and $\mathbb{P}[A_j]=1$ for all $j>1$. I need to show that the probability of the intersection of all those events $A_j$, where j goes from 1 to infinity, is also $1$.

From what I understand, the events we have are not dependent so we can use the formula for a joint probability, so it will be the product of the probabilities of the events. However, I am not sure whether that formula holds in the general case.

Any suggestions?

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  • $\begingroup$ What do you mean by "the general case"? $\endgroup$ – KSmarts Feb 9 '15 at 18:51
  • $\begingroup$ If we have more than two events. $\endgroup$ – madlin Feb 9 '15 at 18:55
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$$\left(\bigcap_j A_{j}\right)^{c}=\bigcup_j A_{j}^{c}$$ So if we deal with a countable intersection then: $$P\left(\bigcap_j A_{j}\right)=1-P\left(\left(\bigcap_j A_{j}\right)^{c}\right)=1-P\left(\bigcup_j A_{j}^{c}\right)\geq$$$$1-\sum_j P\left(A_{j}^{c}\right)=1-\sum_j\left(1-P\left(A_{j}\right)\right)=1$$

Note that independency of events is not used.

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Yes, you can use the formula for joint probability. This will give $$\Bbb{P}\left(\bigcap_{j=1}^{\infty} A_j\right)=\prod_{j=1}^{\infty}\Bbb{P}(A_j)= \prod_{j=1}^{\infty}1=1$$ If you are concerned about using joint probability for more than two events, consider this: we can define new events $B_1, B_2,\ldots$ where $B_1=A_1\wedge A_2$, and in general, $B_n=A_{2n}\wedge A_{2n+1}$. What is the joint probability of the two events $B_1$ and $B_2$?

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