Additional info for coin tossing probability?

Question

My professor is insane. I thought the answer was 1/2 which is rational, but he says it isn't that easy. He used a formula such as Pn (with a hat of the 'P'- no idea what he means by hat) = X1+X2+...Xn-1+ Xn divided by n. He also mentioned "chi square for goodness of fit" Any ideas using either of these? It is known that for a large number of coin tosses, the proportion of tosses which turn up heads is approximately the same as the actual probability of obtaining a head on a single toss.In other words, Number of heads/Total Tosses= P(Head on a single toss.) This proportion is non-deterministic, but by a theorom proved called the Strong Law of large Numbers, the sequence of proportions converges to the value for P(head on a single toss) with probability one. (ie. 100% chance) Is the value for P (head on a single toss) a rational # r an irrational #? Use any method you want in attempting to address this question. There may not be a correct answer, just support your reasoning for the answ

Answer 1

Holy cow, that is a toughy. I saw the answers to the same question that you asked a few hours ago. I think that there is a problem with saying that the probability has to be rational just because

# of heads/# of flips

is rational. You can come up with a sequence of rational numbers that converge to an irrational number. You have to consider that this is a limit here.

My gut tells me that this will be irrational. My reasoning is that you can list the rational numbers between zero and one since that is a countable list. I would think that each of these possbilities have probability 0, which means that the probability of heads would have to be irrational.

But I don't know for sure. This is just a guess. I'm not even sure that makes any sense.

Answer 2

If you ASSUME from the start that the coin is fair, then the probability of getting heads is 1/2, and tails also 1/2.

However, a real coin may or may not be perfectly fair. To determine how fair it is, you need to actually measure it. If you flip the coin 10 times and get 6 heads and 4 tails, is the coin fair? Most people would say this coin is probably fair, because the chance of a fair coin deviating from the average expected result in only 10 flips is quite high.

However, if you flipped the coin 1 million times and got 600,000 heads and 400,000 tails, this coin is almost certainly not fair. Why? Because the probability of a fair coin deviating this far from the mean this consistently is very low. Hence, we can be almost totally sure that the coin is not far, and is biased towards heads.