How do we arrive at the conclusion that P(Head) =0.5 for a fair coin? In Feynman's 'Lectures on Physics', I read a chapter on probability which tells that P(Head) for a fair coin 'approaches' 0.5 as no. of trials that we take goes to infinity (well, I tossed the coin 50 times & got heads 17 times, instead of 25 :-) ...). Can someone elaborate?
 A: It is implied by the law of large numbers - the average sum of i.i.d. random variables (e.g. tosses of the fair coin) goes to the expectation.
A: Everyone's answering this mathematically. I think a better answer is experimental. Andrew Gelman has referred to biased coins as the unicorn of probability theory; see also this paper by Andrew Gelman and Deborah Nolan. The basic idea is that coin tossing is a deterministic process, and the randomness comes from our uncertainty in the initial conditions; half the possible initial conditions lead to heads and half to tails. To bias a coin to come up heads, it would have to slow down in  midair when heads was facing up and speed up when tails is facing up. Unless you have installed some sort of rocket boosters on your coin this is not possible.
A: It is entirely plausible that your coin is not fair. But then again, going back to your little experiment, the probability that a FAIR coin tossed 50 times has 17 heads and 33 tails is FINITE. Which means it can occur. 
A: First of all, keep in your mind that probability is a tool of mathematics. Although you can apply mathematics in the real world, that does not mean that everything true in mathematics is true in the real world as well. This works in the opposite direction as well.
A fair coin is a mathematical abstraction that is defined as a coin that when tossed has a probability of $0.5$ of landing on heads and an equal probability of landing on tails, thus the name "fair". You define it that way and it is automatically true. Building a truly fair coin in the real world would require a ridiculous amount of time and perhaps nanotechnology that we do not have.
So, let's assume that somehow you acquire a real-world fair coin. There is one last requirement to be able to "simulate" probability: an infinite number of experiments. Because that is how you interpret probability: Let $a$ be a sequence that is defined as such: $a_{i} = h/i$ , where $h$ is the number of heads so far and $i$ is the number of experiments. If the coin in this experiment is fair, thus the probability of heads $0.5$, this sequence converges to $0.5$. 
So, as any other sequence, you can interpet this as follows: After a constant number of experiments, the ratio of heads to experiments will be in the "neighborhood" (i.e. very close) to 0.5. The more experiments you conduct, the smaller this neighborhood will be.
