10 Between practical observation and mathematical theory: The empirical law of large numbers.

Slide 0

Welcome to a new episode in which we once again talk about math. This time we will also deal with random experiments and very specifically with what happens when a certain random experiment is conducted a very large number of times.

As is often the case in mathematics, we’re interested in possible regular patterns.

One more thing: We will see some random experiments that we have already used in previous episodes, and we’ll simply look at them in a new way.

Slide 1

And so we immediately begin with them. We roll a completely normal die 10 times, 100 times, and 1,000 times. You have already seen the absolute numbers in the previous episode.

Slide 2

If you enter the data into a bar chart, then you see a very irregular picture for N = 10. The 4 occurred four times, the 1 twice, and all other numbers were each rolled once.

Slide 3

For N = 100 the bar chart already looks somewhat more even. However, the bars are not the same height as you would expect. Clearly, the 2 and 3 were rolled considerably less often than the 5 and 6.

Slide 4

That changes when N = 1,000, that is, when the die is rolled 1,000 times. The 2 caught up significantly, and the 5 lost a bit of ground. But above all it has become clear that the number of rolls for the individual numbers between 1 and 6 gradually grow closer together.

What would you expect for N = 10,000 or N = 100,000? Presumably, the picture would become much more even and the deviations from one-sixth of the respective number of rolls would become smaller.

Slide 5

Let’s look at this again represented another way. For this chart, I allowed a random number generator to roll the die and count how often a 6 turned up. Then the relative frequency was determined, and you recognize that for a small value of N, this frequency varies a lot, but as of N = 1,000 it clearly evens out and lies slightly over the value of 0.15. Theoretically, it would be one-sixth, thus 0.166666… etc. or – stated correctly mathematically because we don’t really like “etc.” – 0.16 repeating.

Slide 6

Here we have now placed a band with a width of 0.1 around this value of 0.1666, thus 0.05 above and 0.05 below.

At the beginning, the relative frequency lies outside of this band, but after a certain number of times, the relative frequency remains within this band. And if we were to make this band narrower, such as half as wide, presumably all values would also lie within this band as of a somewhat larger N. 

Clearly, the empirically obtained relative frequency of rolls with a result of 6 grows closer to the theoretical probability with many experiments.

Slide 7

We have also done this before, namely rolling two dice and adding up the sum of the spots. What’s important here is that we did not just roll 1,000 times, but we also noted down the intermediate results at intervals of 100.

Slide 8

And here are the relative frequencies for selected sums – 2, 3, 5, 7, and 12 – in their evolution from N = 100 to N = 1,000, although I left out a couple of steps to keep the slide manageable.

For the sums of 2 and 3, the relative frequencies fluctuate significantly and are also a bit removed from the theoretical probability. Here the values for N = 100 were not worse than those for N = 1,000.

Slide 9

Things look better for the sums of 5, 7, and 12 and the relative frequencies become closer to the theoretical probabilities and both values are even identical for the sum of 7.

However, to really verify the guess that the empirical results will become closer to the theoretical probability in the long run, we should expand the series of experiments.

Slide 10

By the way, you can see the trend somewhat better if you view the frequency distributions in a bar chart.

Slide 11

Let’s review the two random experiments once again.

For both experiments, it was initially clear that anything is possible for a small number of experiments. No trend can be recognized.

As N becomes larger, however, the empirical probability gradually grows closers to the theoretical probability.

We refer to this phenomenon as the empirical law of large numbers. It states: If you conduct a random experiment a very large number of times, the relative frequencies of the individual events will become closer to a certain value. And of course these events do not have to be equally probable. We saw this indeed in the example of the sum of the spots.

Slide 12

What happens if we don’t know of any theoretical probability? Absolutely nothing. The concept is easily transferable.

As an example, let’s take the thumbtacks that we’re also already familiar with. We tossed it 1,000 times, and it landed on its head 633 times and on its side 367 times. It is certainly not unreasonable to assume that the result in this case will also likely stabilize if we conduct the random experiment a large number of times.

Here as well, the empirical law of large numbers applies.

As the number of experiments increases, the relative frequency of an observed event stabilizes.

Slide 13

That’s all for today. Thank you for being here, and I look forward to seeing you in the next episode.