Does the likelihood of an event increase with the number of times it does not occur?

I would seem logical that the more times an event does not happen, the more likely it is to happen, for example: If a coin is flipped and it lands on tails 10 times in a row it would seam more likely that the next flip will result in heads.

The Infinite Monkey Theorem is one such idea that suggests this is true,
http://en.wikipedia.org/wiki/Infinite_monkey_theorem

It states that if some number of monkeys are left in a room with typewriters for an infinite amount of time then they will eventually compose all written texts ever produced. This seems to suggest that since the chance of the monkeys writing a work, say Shakespeare's Romeo and Juliet, is very low. The more times they do not write it, the more likely they are to write it, until the chance becomes significant and it, the writing of the play, happens.

However another idea, Gambler's Fallacy states quite the opposite.
http://en.wikipedia.org/wiki/Gambler%27s_fallacy

It states that the chance of an event does not increase with the number of times it does not occur.

So what is the answer? Does the likelihood of an event go up the more times it does not happen, or does it stay the same? And if it does stay the same then how does one explain the Infinite Monkey Theorem?

Solution 1:

The Infinite Monkey Theorem does not suggest that the "more times they do not write it, the more likely they are to write it, until the chance becomes significant and it, the writing of the play, happens." Rather, what it says informally is that the longer they have been writing, the more likely they are to have written a given string. The monkey is just as likely to start with the complete works of Shakespeare from keystroke 1 as from keystroke $10^{400,000}$. However, the longer the string of successive keystrokes, the more likely any given substring can be found there. Thus, for example, the complete works of Shakespeare are much more likely to be found in the string of the first $10^{400,000}$ keystrokes than in the string of the first $10^{300,000}$ keystrokes. That's because the former is $10^{100,000}$ times as long.

Solution 2:

The Infinite Monkey Theorem (I didn't know it was a theorem!) basically says that a given finite string of text will appear with probability 1 in an infinite truly random string of text. What it means to be "truly random" is the delicate point.

Anyway, in practice, you cannot produce an infinite string of text "at once" but what you can do (employing monkeys, tossing dice, or instructing your laptop) is to print out a random sequence of letters of increasing, albeit finite, length. As the length increases, so does the probability to find a precise string embedded in that sequence, and this may give the false impression that the chance of producing it improves, because of previous "failures".

In fact it is not so, as the Gambler's Fallacy says. If a perfect coin ("perfect" meaning that "head" has exactly a 50% chance) is tossed ten times and you get "head" ten times, "head" has still a 50% chance at the eleventh toss. Believing the opposite, namely that the chance of getting another "head" is lower than 50%, would be equivalent to believing that the coin has some sort of "internal mechanism" that remembers the past flippings, which is rather absurd.

Solution 3:

There is another example that may help you. Suppose you are waiting a bus. Let us make a simple mathematical model of this situation. Consider that the waiting time is a random variable with an exponential distribution (say of parameter 1). This is a very common model for waiting times of all kinds. However with this model you have the following surprising property:

Suppose you have already been waiting the bus for 5 minutes. Then the probability to wait more than 15 minutes knowing you have already been waiting for 5 minutes is the same as the probability of waiting for 10 minutes. So it is just as if you had just arrived.

This property is called "memoryless" and is a common feature of only two distributions : the exponential and the geometric one. The geometric distribution appears in you coin problem as the law of the time of the first head (first success in Bernoulli trials).

However, if you take another model for the bus waiting time distribution, say for instance a Gamma distribution, then you do not have the memoryless property anymore, meaning that the probability to wait more than 15 minutes knowing you have already been waiting for 5 minutes is not the same as the probability of waiting for another 10 minutes, which might be more in accordance with our intuition somehow.

So, it this example, you see that according to the choice of the randomness model, you can have both answers to you initial question : yes and no, it depends !