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As with any proof, we should start with the statement we are trying to prove, and then start the proof proper. The statement of the weak law of large numbers for a binary variable is the following: Let X_1, X_2, ..., X_n be independent, identically distributed binary variables (in layman's terms: they're coinflips that don't affect each other and have the same probability p). Define Y_n = (X_1 + X_2 + ... + X_n)/n. Then for any epsilon > 0, lim_{n -> infinity} Pr(|Y_n - p| > epsilon) = 0. Writing out the limit: for any delta > 0 and any epsilon > 0, there is some N such that for any n>N, Pr(|Y_n - p| > epsilon) < delta To prove it, we will need a few lemmas. Definition: X and Y are independent if for any outcomes i, j, P(X = i, Y = j) = P(X = i) * P(Y = j). Definition: For a discrete variable X, E(X) = sum_i i*P(X = i) Note the summation in the above. Lemma 1: For any two independent variables X and Y, E(XY) = E(X) E(Y). Proof: E(XY) = sum_{i, j} i*j*P(X = i, Y = j) = sum_{i, j} i*P(x = i) * j * P(Y = j) = (sum_i i*P(x = i)) (sum_j j*P(x = j)) = E(X) E(Y) Lemma 2: Assume X is a variable with all positive outcomes. Then for any a, P(X > a) <= E(X)/a. Proof: E(X) = sum_i i*P(X = i) = sum_{i > a} i*P(X = i) + sum_{i <= a} i*P(X = i) >= sum_{i > a} a*P(X = i) + sum_{i <= a} 0*P(X = i) = a*sum_{i > a} P(X = i) = a*P(X > a), so P(X > a) <= E(X)/a. Lemma 3: If X and Y are independent, then X - a and Y - b are independent. Left to the reader. Lemma 4: E(X - p) = 0. Left to the reader. Lemma 5: E((X - p)^2) = p - p^2. Left to the reader. Lemma 6: For any variables X and Y, E(X + Y) = E(X) + E(Y) (no assumption of independence needed). Left to the reader. Now, as is usual for limit proofs, we work backwards from the statement we want to prove to the statements we can prove. We want to prove that for any delta > 0 and any epsilon > 0, there is some N such that for any n>N, Pr(|Y_n - p| > epsilon) < delta Equivalently, for any delta > 0 and any epsilon > 0, there is some N such that for any n>N, Pr(|(sum(X_i))/n - p| > epsilon) < delta Equivalently, for any delta > 0 and any epsilon > 0, there is some N such that for any n>N, Pr(|sum(X_i) - p*n| > epsilon*n) < delta I want to note here, once again, that this shows what I've been saying: that this is about a range of possibilities. In this case, that range is epsilon*n around the "perfect" outcome. Equivalently, for any delta > 0 and any epsilon > 0, there is some N such that for any n>N, Pr((sum(X_i) - p*n)^2 > epsilon^2*n^2) < delta Equivalently, for any delta > 0 and any epsilon > 0, there is some N such that for any n>N, Pr((sum(X_i - p))^2 > epsilon^2*n^2) < delta Applying lemma 2 (since squares are always positive), we know this is true as long as E((sum(X_i - p))^2) < delta*epsilon^2*n^2, because then Pr((sum(X_i - p))^2 > epsilon^2*n^2) <= E((sum(X_i - p))^2)/(epsilon^2 * n^2) < delta. (sum(X_i - p))^2 = sum_{i, j} (X_i - p)(X_j - p) = sum_i (X_i - p)^2 + sum_{i =/= j} (X_i - p)(X_j - p), so E((sum(X_i - p))^2) = E(sum_i (X_i - p)^2 + sum_{i =/= j} (X_i - p)(X_j - p)) By lemma 6, we can split this sum up into individual terms. The first term is sum_i E((X_i - p)^2) = sum_i (p - p^2) = n*(p - p^2) by lemma 5. The second term is sum_{i =/= j} E((X_i - p)(X_j - p)) = sum_{i =/= j} E(X_i - p) E(X_j - p) by lemma 1, = 0 by lemma 4. So the condition we want is n*(p - p^2) < n^2*delta*epsilon^2, or n > (p - p^2)/(delta*epsilon^2). Which means choose N = ceil((p - p^2)/(delta*epsilon^2)), and the statement follows. This proof generalizes quite easily; all that's necessary is to replace p by E(X_i) and (p - p^2) by E((X_i - E(X_i))^2). I was waiting for you to show any interest in the actual proof, rather than insisting that it hadn't been proven.

Indians are generally considered to be human.

This OP is just a hypothetical discussion initiated by Charon for SFN members and not an active policy. Although I agree with the essence of your thoughts regarding the US establishement and Israel, it is an unnecessary distraction in this discussion.

I posted [math] \lim_{n \rightarrow \infty } |h-t|\rightarrow \infty[/math] meaning the more times you toss a fair coin the larger the probability that there will be a large difference between number of heads and tails. You replied: Your statement is incorrect, I posted the output of a computer program output to highlight that. Do we need to defend mainstream math and science in the mainstream section? Anyway, here is another attempt. This time all combinations are used, not a random selection. As example, lets use a case with a fair coin thrown 32 times. I use 32 as an initial example since it gives a reasonably large number of possible outcomes but not more than could be analysed in full on a regular computer if required. The list below shows how many combinations there exists of the different outcomes. For instance there is only one way to get 32 heads in 32 throws, that is to throw 32 heads in a row. 31 heads and one tail could be thrown in 32 different ways; the single tail could be any one of the 32 throws. heads tails combinations 32 0 1 31 1 32 30 2 496 29 3 4960 28 4 35960 27 5 201376 26 6 906192 25 7 3365856 24 8 10518300 23 9 28048800 22 10 64512240 21 11 129024480 20 12 225792840 19 13 347373600 18 14 471435600 17 15 565722720 16 16 601080390 15 17 565722720 14 18 471435600 13 19 347373600 12 20 225792840 11 21 129024480 10 22 64512240 9 23 28048800 8 24 10518300 7 25 3365856 6 26 906192 5 27 201376 4 28 35960 3 29 4960 2 30 496 1 31 32 0 32 1 4294967296 We see that the most common possible outcome is 16 heads and 16 tails, that is 601080390 of 4294967296. As an example; let's take a closer look at the number of combinations surrounding 16 of each: 19 13 347373600 18 14 471435600 17 15 565722720 16 16 601080390 15 17 565722720 14 18 471435600 13 19 347373600 Each of the 4294967296 combinations have the same probability to be thrown. If we sum the possible outcome of for instance 18 heads, 17 heads, 15 heads and 14 heads we get 2074316640, that is more than the 601080390 possible outcomes that have exactly 16 heads and 16 tails. Checking one of them in more detail: 17 heads 15 tails and 15 heads 17 tails have the same number of possible combinations. So the probability of throwing 17 heads 15 tails or 15 heads 17 tails is the same. That means that on average, when one of the equally probable 17 heads 15 tails or 15 heads 17 tails is thrown, the number of heads will be 16. But |heads-tails| (absolute difference) will be |17-15|=2 or |15-17|=2. Questions: If throwing coin 32 times, counting all possible outcomes what is the probability of getting exactly 16 of each compared to not getting exactly 16 of each? If counting all the possible outcomes, what will it tell us about the probability of throwing heads? If throwing a coin 32 times over and over, what would be the average of |heads-tails| (absolute difference)? Hint: it will not be 0 as you keep claiming. If increasing number of throws how will it affect average of |heads-tails|? Hint: it will not approach zero as you keep claiming. Is the above a feasible approach according to your question regarding using "all combinations"? Should we do some calculations and compare with simulations? I'm not going to put time into a scenario if you are going to reject it anyway, so this post is the first in a possible sequence.

Now I am more inclined to think that The volcanic activity could be the most likely factor there is an alternative theory that our sun might had between 2 to 5 percent more mass in the past but solar winds removed the excess. I remember hearing about it a few years ago but took a bit to find a relevant link https://www.google.com/amp/s/www.space.com/amp/14565-earth-climate-young-sun-paradox.html Anyways googling Young sun paradox will pull up some hits one of the more recent suggested solutions is due to higher solar flare activity https://www.google.com/url?sa=t&source=web&rct=j&url=https://ore.exeter.ac.uk/repository/bitstream/handle/10871/31990/NatGeo_VAirapetian.pdf%3Fsequence%3D1%26isAllowed%3Dy&ved=2ahUKEwjWuuDbmNfkAhXK6Z4KHd8MBEMQFjAEegQIBBAB&usg=AOvVaw2oatKbD466MJR4nUI4sD8o

*Car driving down the road with the driver asleep.* Indian driver: "Crud. I just lost connection." *Car runs over someone.* *Connection comes back* Indian driver: "Ah. There we go." Breaking News: Man arrested after hit and run. Blames Indians.

Buffering...

If I’m forced to choose, I’ll gladly accept some handful of entitled white people complaining over far larger numbers of deserving black people struggling and/or suffering.

Lol! I think raider accidentally switched reference frames there and may have been commenting on self driving vehicles?

Then how do you explain the elevation changes of several of the mountain region's or the impressions that match those of a leftover lake or those that match tributaries as per remnants of rivers ? A continental glacier leaves a completely different pattern after it melts or evaporates https://en.m.wikipedia.org/wiki/List_of_mountains_on_Mars_by_height Here is the list of canyons or Valles https://en.m.wikipedia.org/wiki/List_of_valles_on_Mars Here as well https://en.m.wikipedia.org/wiki/Lakes_on_Mars Many of these have reference papers I suggest you start there

Okay, I understand what you are saying but I don't understand why the Government should get a free pass in that instance. If the government goes to war, we have to pay. If the government spends money on a wall, we have to pay. If the government agrees to fund rebels in Nicaragua, we have to pay. If the government decides to fund Medicaid , we have to pay. But if the government steals from someone, we don't have to pay. In other words we have to pay for all the decisions, good or bad, except for this narrow and ill-defined version of misdeeds. Prosocuter conspires with the police to put an innocent man in jail for 20 years. Tough luck, no compensation. CIA assassinates some poor farmer in Central America because he saw their illegal activity. Too bad, no compensation for their family. Some cop beats you nearly to death because the department hired some psycho and covered up his previous misdeeds over the years. Hope you didn't think the government was going to help with the bills. The Canadian government lies about taxes you paid so they could seize your property and sell it to a developer. Guess you'll just have to start over. Seems like a scary world to me.

Laptops are perfectly natural and subject to evolutionary forces. In the future, they will evolve into Terminators and come back in time to kill you for starting the resistance. Or, they'll evolve into a Machine World, where humans are grown not born, and turn us all into batteries.

Is this car idea serious? A subcompact car* has an interior volume of 2,405–2,830 liters (engine compartment, tires etc excluded) A Passenger cars compact* (PC/C) weights 1100 kg–1360 kg. So a regular car will float if sealed. Your idea is to build a car that weighs about 2 to 3 times more than a standard car does? And then make it fly by using four small propellers? From a commercial point of view; how is the fuel economy for such a vehicle if it is even physically possible? Example as comparison: R44 Police Helicopter has a maximum gross weight 1134 kg**. You propose a vehicle that is 2-3 times heavier than that helicopter and make it fly by using four 17 inch propellers. Second example: A commercial drone fitted with 21-inch propellers weights approximately maximum 15 kg*** at takeoff. *) There are different classifications; examples used to illustrate taken from https://en.wikipedia.org/wiki/Vehicle_size_class **) https://robinsonheli.com/r44-police-helicopter-specifications/ ***) Example: https://www.dji.com/matrice600/info#faq

It is not contradictory, because the range is growing. For 200 flips, the range is between 99 and 101 heads - 3 outcomes. For 2000 flips, it's between 990 and 1010 - 21 outcomes. For 20000 flips, it's between 9900 and 10100 flips - 201 outcomes. For 200 flips, the probability of exactly 100 heads is about 5.6%. For 2000 flips, the probability of exactly 1000 heads is about 1.8%. For 20000 flips, the probability of exactly 10000 heads is about 0.56%. The probability of getting exactly the "perfect" number of heads is decreasing, but the probability of hitting a range including 1/2 is increasing.

You have a balanced chemical equation, telling you how many moles of each reactant relates to moles of product. And you can convert from moles to volume.

Equal opportunities for all is unrealistic as a solution, since we're not all equal, both metaphorically and litterally. INow and Zap and CharonY are suggesting equity for all, which only seems unrealistic, because we're currently afraid it's not economically viable; much like immigration (kinda full circle)... Only room for one crusade per thread.

I'm not sure this is a fair summary of any of our positions, but if we assume for purposes of this discussion that it is, then it's worth noting that creating a society with equal opportunity for all would be a totalitarian nightmare, and we'd have no way of measuring it even if we accept that risk (we can only measure the outcomes). Here's an article I read a while back that helped further open my eyes to some of the involved challenges with the approach. Given your interest, you might enjoy the read, too: https://www.vox.com/2015/9/21/9334215/equality-of-opportunity

We seem to have different visions of the society we want to live in... Seems JC and I favor equal opportunity for all. INow and Zapatos are in favor of equal outcomes for all. And that just isn't going to happen ( because of the multitude of factors involved ). Don't have a clue what Mistermack wants

So you pulled a rabbit out of a hat. This should warrant some kind of round of applause or something?

I really don't know why I would even bother to ask, at this point.

That's a non sequitor. That is interesting, it's something I've been mulling over for a few years now. It seems to make sense, more sense than Mars was warm and balmy at a time when the sun was dim. It would, since it's you that suggested it... I think you mean dimmer and perhaps at a time when Mars was closer to it; it only takes a degree to turn an icy world into a liquid one, and then back again.

Mars was closer? Citation please... Please note, I didn't say Mars was closer than the Earth; and do you really need a citation that one degree stands between ice and water?

Having re-read the title and OP, I'm confused as to what your idea is; it's like you're asking if steve was asleep last night, because I couldn't and I'm far more warm and comfortable.

Americans are full of crap, all this bleeding heart guilt over black Americans, while actively colluding with and supporting the Israelis in their never-ending brutal campaign of ethnic cleansing. Forget the historical atrocities. Stop doing it now, and cut out the blatant hypocrisy.