8 ms·
No need to look at fancy paradoxes, just think about the following. What does it mean that tossing a fair coin has a 50 % probability of showing heads? If you
by danbruc 10y ago
No need to look at fancy paradoxes, just think about the following.
What does it mean that tossing a fair coin has a 50 % probability of showing heads?
If you think you know the answer, you are probably wrong.
EDIT: Instead of just voting this down, try to give an answer. If you think it is easy, you have not thought about it careful enough.
- yellowstuff 10y agoThe frequentist answer is that you could get the proportion of tosses that come up heads arbitrarily close to 50% in a large enough sample. The Bayesian answer is that given all the evidence available to me I would be willing to bet $1 to win $1 if the next toss comes up heads. (Since I work in finance I'll add that this assumes I'm risk neutral, so losing $1 is exactly as bad for me as winning $1 is good.) Except for the risk-neutrality detail this is all Probability 101, right? Or are you thinking of something else?
- danbruc 10y agoThat the relative frequency converges to 50 % is obviously not true. You would have to be exceptionally lucky, but you could get heads, heads, tails repeated for ever and therefore the relative frequency of heads would fluctuate increasingly tiny amounts around 66.(6) %. This of course has probability zero, but it is not impossible. And there are of course many other sequences of outcomes for which the relative frequency does not converge to 50 %. So at best you could say that the relative frequency converges to 50 % with high probability, but now you have a circular definition because you make use of probabilities while defining probabilities. The Bayesian view is problematic for several reasons. Why do I need someone with believes about the coin, we are talking about intrinsic properties of tossing a coin after all. And if that is not enough, we also throw some betting in. Tossing a coin does certainly not depends on the invention of money and gambling, at least ignoring that coins are usually money. Last but not least you have to explain where your believe about a 50 % probability for heads comes from and how it is justified. I could certainly believe that the probability for heads is 25 %, that would not be a good believe.
- p1esk 10y agoSo, what's the right answer?
- danbruc 10y agoIf I knew, I wouldn't have asked. And as far as I can tell nobody knows or at least there is no commonly accepted answer.
- jgehrcke 10y agoI hope to have convinced with my previous answer that the mathematical definition of probability is key to resolving your paradox here. Mathematically (using the concept of infinity) and also physically (using the concept of large numbers) there is a simple and commonly accepted answer to your question. Since that definition is very basic, it is hard to find a direct answer to your question. You might find it interesting that in physics when we deal with large numbers that are not infinity it is usually good enough to know that a certain event is pretty damn unlikely (and it is essential to corroborate this with numbers as in "this happens once in 10^100 years"). In physics, we rarely have 100 % certainty. Logical consistency and a certainty that is large enough are usually key to success (i.e. growth of knowledge) in natural sciences.
- prmph 10y agoI guess another way of stating the problem is this: say you toss a coin and get heads the first 1000 times. Now you are likely to believe that the coin is biased. But that cannot be proved. It may be that you have simply not tossed the coin enough times to perceive its fairness. Maybe after the 1000th toss you start to get enough tails such that after the millionth toss, it is not at all clear that there is any bias. So practically we can make a judgement as to whether the coin is biased on not based on how many tosses we think is sufficient, but theoretically it is impossible to distinguish a biased coin from a fair one if we toss to infinity
- 10y ago
- b_emery 10y agoIt means that there is a 100% chance that one side of the coin has a head on it.
- danbruc 10y agoWell, that is already in the setup, there is also a 100 % chance that the other side is tails. I don't think that observation will turn out to be very useful.
- llasram 10y agoThat we're modeling the outcome of a coin toss as a sample space set containing at least two event elements, one of which we call "heads," and that we have a probability measure which assigns 0.5 to the subset consisting of only the "heads" event.
- danbruc 10y agoYou are presupposing an understanding of probability, aren't you? What does it mean that a probability measure assigns 0.5 to heads?
- criddell 10y agoIf it's a fair coin, then by definition there's a 50% probability of showing heads.
- danbruc 10y agoYes, but what are the consequences of this definition? What does a probability of 50 % for heads imply about the coin? Or what is the difference to an unfair coin with a 40 % probability for heads and a 60 % probability for tails?
- criddell 10y agoI think you're answering your own question.
- Retra 10y agoThe probability has no implications about the coin. It has implications about how you make decisions concerning the flipping of coins.
- criddell 10y agoIt almost sounds to me as if he is pushing into gambler's fallacy space. Like if a (fair) roulette wheel hasn't hit 7 in 200 spins, then 7 is somehow due.
- TekMol 10y agoYou can play this game for every statement. What does it mean that an elephant is bigger then a dog? If you think you know the answer, you are probably wrong.
- danbruc 10y agoThat one is easy. First let's clarify what exactly we mean with bigger. Let's say we mean larger in volume. Now we pick a procedure for determining the volume of animals, say we submerge them briefly in a large water tank and mark the resulting water level. That animal that lead to the highest water level is the bigger one. Any objections?
- saghm 10y agoBut similar to your objection about the Bayesian view of probability requiring money and betting to be defined, you've now defined the idea of size to be dependent on giant tanks of water
- TekMol 10y agoAny objections? The same objections you have regarding the fair coin. You say you submerge the animals in a tank of water. How does that work? You have this giant tank, put in the animal and one molecule of water? Just like a single toin coss, a single molecule of water will not tell you much. You need hundreds of billions of water molecules? Ok. And how do you know these will behave in the expected way? Because you have a model in your head how molecules behave? Well, I have a model how fair coins behave. Because you saw different sized objects result in different water levels before? I saw fair and unfair coins result in different head:tail distributions before.