Probability of occuring of An infinitely often

Question

IF probability of occurance of A , infinitely often is 0, does it mean probability of occurance of A complement , infinitely often is 1???

Answer 1

No it doesn't.

If you have a finite number of trials then neither A nor complement A will happen an infinite number of times.

If you have an infinite number of trials then one or both of A and complement A must occur an infinite number of times. Furthermore if the probability of A remains constant then the only possibilities are:

• A never occurs, complement A always occurs.
• A always occurs, complement A never occurs.
• Both A and complement A occur an infinite number of times.

Answer 2

The opposite of A occurring infinitely often is not

'not A' occurring infinitely often

but

'A' not occurring infinitely often

i.e.

'A' occurring only a finite number of times.

As Mark pointed out, the meaning of this for not A depends on whether you have a finite number of trials ('occurrences') or not:

• finite number of trials: also not A can only occur a finite number of times

• infinite number of trials: A only occurs for a finite number of trials, so not A occurs for the rest of the trials which are infinitely many (assuming only A and not A can occur in each trial).

Answer 3

This is interesting.

Try instead a finite number of trials.

We do 100 trials where the outcomes are A or A'. If the the probability of getting any As is zero then it follows that the probability of getting 100 A' is 1.

Extend to 1000 trials ... 1000(A) probability 0, 1000(a') probability 1.

Extend to infinity ...

How surprising, yes it appears that the answer is "yes".