Certainty and impossibility.

Something that just crossed my mind: If a certain event has a (likely very small) possibility of occurring every so often, would it be technically correct to say it is a mathematical certainty that the event would eventually occur?

For instance, if numbers were somehow constantly drawn in a truly random manner, is it theoretically possible that a certain number would never be drawn, ever?

Re: Certainty and impossibility.

Quote:

Originally Posted by

**Pulsys** Something that just crossed my mind: If a certain event has a (likely very small) possibility of occurring every so often, would it be technically correct to say it is a mathematical certainty that the event would eventually occur?

For instance, if numbers were somehow constantly drawn in a truly random manner, is it theoretically possible that a certain number would never be drawn, ever?

Yes. No.

See Infinite monkey theorem - Wikipedia, the free encyclopedia

Re: Certainty and impossibility.

Quote:

Originally Posted by

**Pulsys** For instance, if numbers were somehow constantly drawn in a truly random manner, is it theoretically possible that a certain number would never be drawn, ever?

Indeed. Consider rolling a fair die, then $\displaystyle P(6) = \dfrac{1}{6}$ and then $\displaystyle P(n sixes) = \dfrac{1}{6^n}$

Re: Certainty and impossibility.

It's probably important to distinguish between an event that has probability 0 and one that small but positive probability. If you are drawing from a uniform distribution on [0, 1], then it is possible that you will draw 1/2, but the probability is 0 that this will ever occur, even if you take infinitely many samples. Any positive probability on 1/2, though, will make us draw 1/2 eventually.

Some odd things can happen even in the discrete case, if you are willing to bend the rules a little bit. Suppose we are flipping coins forever. Well, if the coin is fair we will flip a tails eventually, with probability 1 - it is "possible" in some respect to never flip a tails, but this event has probability 0. If we assume instead, however, that the probability of flipping a tails is $\displaystyle {(j+1)^{-2}}}$ on the j'th flip, things get more interesting. With probability exactly 1/2, we will never flip a tails ever; moreover, with probability 1, we will eventually reach a point where we flip nothing but heads. If we change this slightly to a probability of $\displaystyle (j+1)^{-1}$, the game changes; the probability of us flipping no tails is 0, and in fact we will be flipping infinitely many of them with probability 1.