For example, when I flip a coin, I have a 1/2 chance of getting head, and the same chance at getting tails. With that theory, if I toss a coin 50 times, I should get 25 heads, and 25 tails.
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However, I did 3 rounds of 50 coin flips, with results of:
1. 28 heads, 22 tails
2. 27 heads, 23 tails
3. 27 heads, 23 tails.
I assumed that perhaps the coins weren’t “true”. Maybe the sides are waited different, maybe I tossed with different height/force each time. So I went virtual, and switched to having a computer roll a dice.
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I should have a 1/6 chance at rolling a number between 1-6. So 60 rolls, should have each number come up 10 times. But in practice my results were:
1. 14
2. 9
3. 8
4. 13
5. 6
6. 10
So how come practice/reality, doesn’t align with theory, even when we take human error out of the equation?
In: 34
>I should get 25 head, and 25 tails.
No, you “should” not, since that’s implying that the many other reasonable outcomes, like 24h 26t or 26h 24t are *impossible outcomes*. They clearly aren’t, it’s just that 25h 25t is the most *probable outcome* and that’s a completely different thing.
In this case it happens because the sample size you’ve investigated is still quite small, especially compared to infinity! According to probability theorems and inequalities like the [Law of Large Numbers](https://en.m.wikipedia.org/wiki/Law_of_large_numbers), the [Glivenko–Cantelli theorem](https://en.m.wikipedia.org/wiki/Glivenko%E2%80%93Cantelli_theorem) and the [Dvoretzky–Kiefer–Wolfowitz–Massart inequality](https://en.m.wikipedia.org/wiki/Dvoretzky%E2%80%93Kiefer%E2%80%93Wolfowitz_inequality) we do have correspondence between the theoretical probability and the observed or empirical probability as the sample size goes to infinity. We also know how the difference between them “behaves along the way” as the sample size goes towards infinity.
Let us take the weak form of the [Law of Large Numbers](https://en.m.wikipedia.org/wiki/Law_of_large_numbers) that’s basically saying that you can get arbitrarily close to the 50/50 proportion you’re expecting in the coin toss case with a “large enough” sample size. Or a bit more precise, you can almost certainly get within a small distance from the true proportion 50/50 as the sample size goes to infinity, for arbitrarily small distances larger than zero.
In summary, we “know” that we’re getting close to the ground truth, that is the theoretical probability, as the sample size increases and we “know” that we’d get to the ground truth exactly if we could take an infinitely large sample (which we can’t in reality but that’s beside the point). This is true in the trivial case of coin flips and sample proportions but some of these guarantees don’t hold in more complicated situations.
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