# Borel Cantelli lemma Question

1. Jun 23, 2012

### calios

guys im 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?

this correct?
or i made a mistake ? thank you

2. Jun 23, 2012

### micromass

Staff Emeritus
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. Jun 23, 2012

### calios

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 happend right?

Last edited: Jun 23, 2012