Does Borel Cantelli lemma apply to infinite universes?

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In summary, Borel-Cantelli lemma 2 states that if a sequence of independent events En has a sum of probabilities that diverges to infinity, then the probability that infinitely many of them occur is 1. This applies to both the event A and its complement, "not A." However, it does not necessarily mean that all En must occur, as demonstrated by the example of the sequence 0,1,0,1,0,1,... in which infinitely many times 0 occurs but is also infinitely times in the complement of 0. Therefore, in an infinite universe, both the event A and its complement can occur infinitely many times.
  • #1
calios
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guys I am bit confused with the statement borel cantelli lemma 2 :
"If the events En are independent and the sum of the probabilities of the En diverges to infinity, then the probability that infinitely many of them occur is 1"

this state if probability of A occur > 0 .
and if the probabilities of A diverges to infinity, then the probability that infinitely many of A occur is 1

the problem is
if probability of opposite A, event[not A] > 0
and if the probabilities of "not A" diverges to infinity
then the probability that infinetely many of [not A] occur is 1.
in other way the probability of A doest occur is 1 too? :confused:

this correct? :confused:
or i made a mistake ? thank you
 
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  • #2
Borel-Cantelli says that the probability that infinitely many of the En occur is 1. It doesn't say that all the En must occur.

Likewise, it might be that infinitely many of the Enc occur. It's not said that they must all occur.

For example, the sequence 0,1,0,1,0,1,... becomes infinitely many times 0 and is infinitely times in the complement of 0.
 
  • #3
micromass said:
Borel-Cantelli says that the probability that infinitely many of the En occur is 1. It doesn't say that all the En must occur.

Likewise, it might be that infinitely many of the Enc occur. It's not said that they must all occur.

For example, the sequence 0,1,0,1,0,1,... becomes infinitely many times 0 and is infinitely times in the complement of 0.

oke u mean..
ley say we have infinite many universe,
and probability Earth occur is 1/6, the probability Earth does not occur is 5/6
so the probability of infinite many Earth occur is 1. and the probability of infinite many Earth does not occur is 1 too.
in infinite universe both are happened right?
 
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1. What is the Borel Cantelli lemma?

The Borel Cantelli lemma is a fundamental theorem in probability theory that describes the convergence of events in a sequence of independent events. It states that if the sum of the probabilities of the events is finite, then the probability of infinitely many events occurring is zero.

2. How is the Borel Cantelli lemma used in probability theory?

The Borel Cantelli lemma is used to determine the convergence of a sequence of events in a probability space. It is often applied in problems involving independent events, such as in the study of random processes and stochastic convergence.

3. Can you provide an example of the Borel Cantelli lemma in action?

One example of the Borel Cantelli lemma is the "three sigma rule" in statistics, where it is used to determine the probability of a data point being outside of three standard deviations from the mean in a normal distribution. This can be applied to determine the likelihood of extreme events occurring in a given dataset.

4. What is the difference between the Borel Cantelli lemma and the Strong Law of Large Numbers?

The Borel Cantelli lemma and the Strong Law of Large Numbers are both important theorems in probability theory, but they have different applications. The Borel Cantelli lemma is used to determine the convergence of a sequence of events, while the Strong Law of Large Numbers is used to describe the behavior of a sequence of random variables as the number of trials increases.

5. Are there any limitations to the Borel Cantelli lemma?

Yes, the Borel Cantelli lemma has some limitations. It only applies to independent events, and it does not guarantee the convergence of a sequence of events in all cases. There are also variations of the Borel Cantelli lemma that are used for specific types of events, such as the Borel Cantelli lemma for martingales.

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