Law of large numbers and probabilities

as the number of trials increases, the empirical probability will get closer and closer to the theoretical probability. This is what the law of large numbers states.

So are there any cases where empirical probability is greater than theoretical one during large trials?

I am not talking about 10 or 100, but million of tries.

Sure. Sampling deviations can go either way, right? If you’re flipping a fair coin a million times, you could get 500,017 heads or 499,063 heads or even bigger deviations in either direction. Of course those numbers may not be a very likely in a million iterations. What ends up being “large” (as in “Large Numbers”) depends on the nature of your experiment. Remember that in mathematics, “large” can get pretty far outside the bounds of our normal physical intuitions.

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Hi @tbhaxor !

Let’s look at an example of the law of large numbers using a “fair” 6 sided dice.

In theory when we roll the dice we have an equal probability of rolling all numbers.

But let’s say we do 12 rolls and we get this:

This is a possible outcome of 12 rolls but it doesn’t seem to match what we expect. An equal distribution of dice values.

Is our imagined dice weighted?
Possibly but let’s try 120000 ish trials but keep the difference between the categories by adding 20000 to each outcome.

When we look at the plot we see something that is much closer to what we expect which is about equal probability for each dice value.

Now if we scaled up and saw the same ratio between the dice values we might suspect that the dice might not be “fair” and our theory for this dice might not match reality.

I hope this clears things up!


You’re talking about probability. Absolutely any outcome is possible, though perhaps not very likely.

Yes makes sense. Also thats why we say it is close to theoretical probability.