# why does theoretical probability not align with practice?

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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:

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 There really isn’t any reason we should expect the results to exactly match the probability, because the probability addresses *each individual roll* and the individual rolls don’t care about the other rolls…in other words, the rolls don’t “know” that they’re “supposed to” work out to 1/6 overall. It’s a matter of scale. The more you repeat the same experiment, the closer to the true distribution you get. Keep flipping that coin. Do it 1000 times. Then, ask yourself (other than “is this good for my wrist”), has the distribution of heads/tails converged over the sequence. It’s the same with the dice Probability/Statistics-wise you are using a very low sample set.

Over a set of say 100,000,000 the numbers will be closer to 50/50 (most likely).

Probability is a matter of chance though. It doesn’t dictate results. The coin has a ~50% chance of either heads or tails, for each toss. None of the tosses are linked. The results of a prior toss do not effect the chances of subsequent tosses. It is also possible that you could throw 32 tails in a row. It’s a 1 in 4,294,967,296 chance though. The more throws the smaller that chance gets. Maybe at the exact same time I was also conducting 3 rounds of 50 coin flips and got the exact opposite results!

Also it might just be a matter of sample size. 50 flips is a lot, but maybe you should try 100 or even 1 thousand. I suspect the more you do it, the closer it is going to get to theoretical probability. >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.