Conjurer

That isn't one of my concerns. I accept that as being a likely scenario. My concern was that if you add or multiply probabilities together to find the most likely outcome over an increasing number of series of trials, you wouldn't get something that represents anything close to what you should arrive at from the law of large numbers, which is approximately half of them being heads or tails.

You should hold on to that thought right there. I think you almost got to the point I was trying to say. Then forget about what else you may I thought I meant by it, because I didn't. That is basically what I have been trying to explain this entire time. There is no proof of probabilities being accurate, because there is no connection between them and the law of large numbers. You cannot add or multiply a series of them in such a way where you get the probability of an event occurring one single time.

Where does this have the probabilities of events occurring in a row in it?

This is enough to put someone into an insane asylum, if you don't have any reference for it. I see no reason to take your word for it. Where is this proof you had for centuries?

The probability of getting heads or tails is 1/2! P(H or T) = 1/2 You have a 1 out of two possible outcomes to get either a heads or a tails. The numerator is the number of outcomes for a success, and the denominator is the number of possible outcomes. What you are asking for doesn't seem to make sense. I don't even think the mathematical tools to even describe what you are asking has even been discovered yet. You are just asking me to discover this. I don't know how you could incorporate a probability into the law of large numbers to show that. I gave you my best guess, but maybe you would have to subtract the probability from an average probability instead...

That is the proper notation of the probability of getting a heads or tails. That should be one of the first things you learn about probability... It shows where you are having this problem of it just looking like I am quasi-randomly putting formula together... I am not looking for a fruitful discussion about it. I am looking for answers. I don't think the law of large numbers has been proven from the basis of considering the probabilities of events occurring in a row. I am wondering if I have already answered my own question, but I don't understand why n^r doesn't seem to work with the coin flip problem. It seems like the permutations with replacement should be r^n. Then you could get 2^(#flips), which would give you the total possible number of outcomes. If my math was correct, then 1/(r!) could be the average probability. Then you could say that the average probability of getting heads or tails would just be 1/(2!)=1/2. Then n^r didn't seem to work in this example...

(n!/(n^r (r! (n - r)!)) - 1/r!) < P(H or T) < (n!/(n^r (r! (n - r)!)) + 1/r!)

the proportion of heads after n flips will almost surely converge to 1/2 as n approaches infinity. 990/2000=0.495 1010/2000=0.505 0.495<x<0.505

No, it almost surely converges to 1/2, which means that it is guaranteed to approach a 1/2 probability or it has a probability of 1 of approaching 1/2. You don't ever seem to be able to grasp anything while only dealing with perfect outcomes. How could you be capable of understanding something more complex than that to set up? n=2000 990<X_n<1010 X_n=(1/2000)(X_1+...+X_n)

https://en.wikipedia.org/wiki/Law_of_large_numbers For example, a fair coin toss is a Bernoulli trial. When a fair coin is flipped once, the theoretical probability that the outcome will be heads is equal to 1/2. Therefore, according to the law of large numbers, the proportion of heads in a "large" number of coin flips "should be" roughly 1/2. In particular, the proportion of heads after n flips will almost surely converge to 1/2 as n approaches infinity. Is the coin going to start landing on its side or something? I don't see why you are so caught up in this. When I did my calculation earlier I just used the variables that could represent any possible range of the number, including the ones that were not even close to the expected value. The average of all the combinations per permutations was 1/r! for the entire range of possible outcomes of any replaceable event as it approached an infinite number of trials.

The law of large numbers By definition, a larger number of flips should approach the expected value or probability of getting heads or tails, which is 1/2

No, it approaches the expected value which is 1/2 You would have to assign values to heads or tails, so that the number of flips is twice the number of possible outcomes for a single flip.

I think we should start over. The law of large numbers is just showing how someone would get the expected value from an experiment. They would add all the outcomes and take the average to determine the expected value. I am saying that it is not possible to use the probabilities of a series of heads or tails to obtain the expected value, or the answer you would get from the law of large numbers. Say I was going to use the law of large numbers, I would assign 0 to heads and 1 to tails. I got 100 heads and 101 tails after a total of 201 flips. I add all of the X_n to get 101, since all the heads add up to zero. I then divide that by 201. Then I get approximately 1/2 Then by that experiment, I proved that the probability of the coin is 1/2 and it is a fair coin. Now I want to calculate what is the probability of me getting 100 head and 101 tails. It comes out to some ridiculously low chance that will ever happen...

I don't think it or the third does really. I thought it was a 1 with a decimal in front of it, but it was actually a comma. The average of heads/tails=1. It makes it to where it only proves what the programmer was capable of coding to simulate randomness where it actually didn't exist. If he/she went to these forums and asked a public opinion on how that program should act, then assuredly, yes, it would make it invalid. I don't it is possible to add probabilities and take an average to get the expected value. In the example, they just used the variation of the data to calculate something using the law of large numbers. Then it is just an average of the date values, not the actual probabilities themselves that are involved in the equation. You appear to be under the impression that what I am saying is impossible, because you have a computer spitting out possible outcomes.

It has been proven experimentally, but not mathematically. I am sure you would have no idea about a basic math concept. I don't understand why you are trying to help me with math, when you cannot even take a basic average to see that the numbers do not add up.

That's what I would have expected, because a random number generator simulates true randomness. Then there is no such thing as true randomness that can be coded into a computer, as far as we know. Possibly, a quantum computer could generate true randomness. I don't know if they have programmed one to do that yet. That has more to do with the basis in how a quantum computer gains it's bits. I guess I should restate my problem, my difficulty with accepting probability theory, is based on: The law of large numbers has not been been proven based on the probabilities of all the possible outcomes. I state this because, in the example of a die roll, it doesn't use the probabilities of getting a certain die roll, which would be 1/6 for each side. Then it says X_1=X_2=... I don't think X_n can stand for a probability. If it can, I don't see how it could or how someone would show any example where it does use probabilities to obtain the expected value from an increasing number of outcomes.

https://en.wikipedia.org/wiki/Law_of_large_numbers probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.

By definition, it is the law of large numbers, but I don't think that is going to be able to help you grasp the concept any more that what statements I have already provided here for you. The problem now, is that That does not agree with the law of large numbers... The problem now is that, the law of large numbers doesn't consider the probabilities of the possible outcomes.

The integral is the same as an undefined interval of sums. The combination formula is not limited to any specific values. In the example the wiki gives about a die roll, it just adds 1-6 and divides by 6 to get 3.5. Then it explains that a large number of roles should average 3.5. I think you are mixing up what the word large implies here. It is just saying that a large number of roles should average to be the expected value. You didn't define the limitations of a and b so then it is the same as the integral. Instead of nCi, instead, I put in the whole combination formula with it being over 2^n.

The fundamental theorem of calculus I don't think it actually matters how large the numbers are being used with it. Any law or theorem should work the same no matter what size variables are used. I just put in what you gave me into wolfram to solve it. That was the combinations divided by the total number of outcomes as the number of flips approach infinity. It gave me the answer that there was 0 chance that I could flip a coin an infinite number of times and get the same amount of heads and tails.

Because the law of calculus says that is the same as adding them all together I used the weak law to calculate it I want the generalized form for any type of event

I thought it could be done both ways. 2^4 = 16 Then that is 4 flips of either heads or tails. Then if it was n^r, then it would be 4^2 = 16 2^6 = 64 That is 6 flips of either heads or tails. Then if it was n^r, then it would be 6^3 = 216 Maybe, you are on to something there. I just checked the first case before. I don't see why n^r wouldn't work in this situation. That is the equation that was given in the middle of the page here. https://www.calculator.net/permutation-and-combination-calculator.html I thought it would just put it into a more generalized form which would be able to work for all cases, in order to be considered a proof. A mathematical proof doesn't seem to be as good if it only works for one special case. I was just using this example to try to get to it. I am not sure how that typo occurred, but I corrected it. I thought it was interesting that the law of large numbers would say that it had an average probability of getting exactly the same out of heads and tails was 1/2 for only 4 flips, when the probability of getting that is 3/8 of all the possible outcomes. lim_(n->∞) 1/n integral(n!)/(2^n (r! (n - r)!)) dn = 0

I took the equation for the number of combinations nCr = (n!)/(r! (n - r)!), and I divided that by the number of permutations with replacement nPr = n^r in order to get the probability of a desirable outcome out of the total number of outcomes. Then I took the integral of that to add up all of the probabilities of those outcomes. Then you say that the law of large numbers says that they all have to be divided by the number of flips, so then I divided that by n. I didn't get one as an answer doing that. It should be 1/r! in any case this happens. If there where 4 flips it would be 1/2! = 1/2. Then if there were 6 flips, it would be 1/3! = 1/6, because r=n/2 in this example.

I added up all the probabilities of getting the same number of heads and tails as the number of flips approach infinity then divided by the number of flips, and I got 1/r! lim_(n->∞) ( integral(n!)/(n^r (r! (n - r)!)) dn)/n = 1/(r!)

How would you propose what range of outcomes should be selected in order to obtain an average of the expected value? I don't know of any established process of doing such a thing in mathematics.