# What happens if you flip an infinite heads and tails coin?

Tags:
1. Jan 6, 2015

### Tungamirai

What happens if you flip a coin with infinite heads and infinite tails? I am not sure if this is the right place to post this question, or if my question even makes sense! just thought about it after reading about the simulation argument

2. Jan 6, 2015

### Doug Huffman

Your question does not make sense. A single fair coin has one obverse and one reverse. There are not an infinity of countable anything. Frequentism is at the root of the failure of conventional statistics, particularly in describing our singular universe. Look into Bayesian statistics.

3. Jan 7, 2015

### chiro

Bayesian statistics just involves conditional probabilities - I'm not quite sure with its relevance to the original question.

You should look at probability theory to understand what constraints a space has to have for it to be a probability space.

If you flip a coin that has two outcomes (heads and tails) infinitely then then the population distribution will be represented by the relative frequencies of the coin toss and the parameter will be a function of said distribution. The infinite part is what matters here as it defines the population distribution and hence the population parameter.

It does not contain any conditional information however - just the zeroth order distribution for a coin toss stochastic process with a parameter p.

4. Jan 7, 2015

### Staff: Mentor

Are you really asking about the effect of population size( number of coin flips)?

5. Jan 7, 2015

### DarthMatter

First you have to know what infinite means. $\frac{\infty}{\infty}$ can be anything. Are there 100 times as many heads as tails? Or are there exactly as many heads as tails?

6. Jan 7, 2015

### FactChecker

Is there a "simulation argument" that could help us understand specifically what your question is?

There is one issue that needs to be clarified: There is an important difference between the odds of a fair coin giving a particular exact sequence of heads and tails versus the believability, given particular sequence result, that the coin really was fair.
1) For a fair coin, the sequence of all heads is no less likely than any other exact sequence of heads and tails.
2) There are aspects of the all-heads sequence that are not at all like a random process. So as heads keeps coming up, it becomes certain that the coin is not fair.
3) Other exact sequences may prove that a coin is not fair. An infinitely long sequence of exactly alternating heads and tails is not from a fair coin.
4) On the other hand, there are randomly mixed heads/tails sequences that have exactly half heads that may be from a fair coin. As the coin tosses go on to infinity, the odds of exactly half being heads goes to zero, but 50/50 remains the most likely value. All heads (or all tails) are always the least likely.

7. Jan 8, 2015

### lavinia

I didn't understand your question before. You are asking what the result of infinitely many flips of a coin will be?

If that is what you meant, then The Strong Law of Large Numbers says that the sample average will converge almost surely to the expectation. So if the coin is fair, an infinite sequence will have an average of 1/2 almost surely - with probability 1. Is that what you were asking?

Last edited: Jan 8, 2015