Past coin tosses affect the latest one if you know about them? [duplicate]

If, as you said, it's an unbiased and fair coin, then it has an equal chance of coming up heads as tails on the 100th flip, by definition, because that's what an unbiased fair coin is: it is a coin that has an equal chance to come up heads or tails on any flip.

On the other hand, if you see a coin come up tails 99 times in a row you had best revisit your assumption that it is an unbiased fair coin. It would be foolish to bet on it coming up heads after that.

Addendum: Here's the reference you asked for: Gambler's fallacy

The restriction is that the coin is fair. There does not seem to be a restriction that the flipping is fair.

With practice I used to be able to get the same result from coin flips - mostly heads or mostly tails. One needs to start with the coin in the smae head up or tail up position, but it is not difficult to make the flipping unfair.

To elaborate on Mark's answer. Let's say you don't necessarily believe that the coin is unbiased, but instead believe it has probability $p$ of coming up heads.

You degree of belief in the value of $p$ can be specified by providing the parameters $a$ and $b$ of a beta distribution. Intuitively, they correspond to the number of times you've seen heads and tails come up already. A typical 'default belief' is given by $a=b=1$, which essentially expresses maximum ignorance - you believe that the true value of $p$ is distributed uniformly between 0 and 1.

The benefit of doing this is that there is a simple method of updating your belief about the distribution of $p$ whenever you see a new coin toss - you simply increment $a$ by 1 whenever you see a head, and increment $b$ by one whenever you see a tail.

The mean of the distribution is simply $a/(a+b)$ and its variance is $ab/[(a+b)^2(a+b+1)]$. If you see 99 tails in a row, then your new values of $a$ and $b$ are $a=1$ and $b=100$, giving an expected value for $p$ of

$$E(p) \approx 0.01$$

$$\mathrm{StDev}(p) \approx 0.01 $$

So with a very high degree of certainty, you believe that the true value of $p$ is 0.01, and certainly isn't much more than 0.04. You would therefore be very naive to bet on heads coming up on the next toss (unless you were given very favorable odds!)