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I am going through a basic course in stochastic processes from the book by Sheldon Ross. In the first chapter it introduces concepts of probability theory and after few fundamental prepositions it gets in to the Borel Cantelli lemma.

Let us consider an event where I have to choose a real number between $0$ and $1$. Suppose I have an infinite number of friends and I ask them to find out the subset to which the number belongs to. Each guess is an event and each collection comprises of numbers which are outcomes. Now if I assign a probability to all the possible events, I sum them up then on obtaining a finite number it can be ascertained that the probability of the union of all the subsets being the correct subset is zero.

Now let us consider that my first friend guesses that it belongs to $[0,\frac {1}{2}]$, my second friend guesses that it belongs to $[0, \frac{1}{4}]$, and so on. In this way my $n^{\text{th}}$ friend guesses that the number belongs to $[0, \frac{1}{2^n}]$. Now the probability of first friend being correct is $\frac{1}{2}$, that of second friend is $\frac{1}{4}$ and so on. If I sum up all these probabilities I obtain a finite number that is the sum of a geometric progression. However the collection of all these numbers is $[0, \frac{1}{2}]$ whose probability of being correct is $\frac{1}{2}$.

Does this breach the theorem or there are some finer details that I have missed? Along with this I would also like to know what makes this lemma a basic lemma in probability. Certainly it doesn't make any sense to me that suddenly a mathematician can consider about events with probabilities summing up to a finite number and their possibility of occurrence. Hope I can learn what cases directed mathematicians to work up to this lemma.

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You should format your question for better readability! –  user3001408 May 26 at 17:01
    
@user3001408 Could you tell me in what way should I format ? –  Primeczar May 27 at 8:20
    
I did a bit! Or you can refer to other questions and see how they are formatted. This is a good practice, a good readability question (especially long ones like this one) is good as people can complete reading and try addressing it. –  user3001408 May 27 at 8:42

2 Answers 2

This does not contradict the Borel-Cantelli Lemma.

Let $E_n$ be the event that the chosen number is less than or equal to $2^{-n}$. As you observed, $\sum_{n=1}^{\infty}Pr(E_n) = \sum_{n=1}^{\infty}2^{-n} = 1 < \infty$. The Borel-Cantelli Lemma implies that the probability that infinitely many of these events occurs is zero. That is, the probability the chosen number is less than or equal to $2^{-n}$ for infinitely many $n$ is zero. This is true because the only $x \in [0, 1]$ such that $x \leq 2^{-n}$ for infinitely many $n$ is $x = 0$, and the probability of the chosen number being $0$ is zero.

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You did not actually specify which event you want to calculate. But let us assume you draw the number from the uniform distribution on $[0,1]$. Then you have the specified events. The Borell-Cantelli lemma is a statement about the event that the value you have chosen lies in infinitely many of the sets your friends chose.

Now, if $x$ lies in infinitely many of the sets $[0, 1/2^n]$, then $x$ must be $0$! And the probability that you pick exactly $0$ is indeed $0$, so the Borel-Cantelli Lemma does apply, since $1/2+1/4+1/8+\ldots=1<\infty$.

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