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Fundamentals of statistics and probability: delightful dice.

1 Fundamentals of statistics and probability: delightful dice.

Slide 0

Math is an engaging topic and must be covered ... of course. This is particularly true for stochastics, which is increasingly significant. Knowledge of stochastics is useful if you want to better understand the changing world around us. So let’s begin. Among other things, this has to do with marvelous dice. Allow yourself to be amazed.

Slide 1

To be more specific, today we’re dealing with statistics, a branch of mathematics that is all about data. How can we collect data? Why do we collect data? Do regular patterns exist that we can exploit? 

Slide 2

We have just mentioned that statistics is about handling data. Above all, however, it’s about handling data correctly.

Let’s look at an example. The class representative is to be elected in a class. Ama, Lin, Noah, and Kofi have run for the position; you see the results on the slide. Even for young students it is clear that Ama will be the new class representative because she received the most votes. Kofi came in second and will be her deputy. Any other result would not be the correct handling of data and would not be accepted by the class.

What do we learn from this? Quite simply, statistics in school can build upon everyday examples in this context.

Slide 3

The next example is also simple and is based on the collection of data. The basis is a survey on how the students get to school. It’s very clear that at this school, most students take the bus, the bicycle comes in second, the car is in third place, and the fewest number of students walk to school.

Here we can see how useful a good representation of data is. In the pie chart shown here, although you cannot see the individual tallies anymore, at a glance you can see that bicycles and buses make up a very high share of the listed transportation options.   

Slide 4

But what do we need data for anyway? For example, to make decisions on a sound basis in more or less difficult situations. You could advise the school involved on the last slide to provide bike racks, based on the data.

Let’s look at another example and ask which means of transportation are actually safe.

Slide 5

In this case, we can obtain a suitable database from the German Federal Statistical Office. Based on the data, we created a table that provides information on traffic deaths in 2018. Obviously, most of the deceased were traveling by car, but the counts are also high for people riding motorcycles 50 cc and larger, simply walking, and riding bicycles. You immediately see that buses, trains, and trams are apparently very safe means of transportation.

Naturally, it would be useful in this case to know how many people use these means of transportation, to compare absolute and relative figures. We’ll leave that out of the equation for now – to keep from getting too complicated right away.

Slide 6

Many possibilities exist for collecting data. They also include observation. If you would like to know whether an intersection is overcrowded, it makes sense to count how many vehicles are waiting at the intersection during a traffic light cycle and how many vehicles make it through the intersection.

Slide 7

Heard this all before? Sounds as tempting as yesterday’s coffee? Sure, we can also use cold coffee to collect data.

And I’ve done just that. On the right side of the chart, you can see what happens if you let hot coffee cool down for about 70 minutes. It goes from around 80 degrees Celsius down to about 30 degrees. But I would like to have you participate in the process.

Slide 7a

That is the cup of coffee. That is the thermometer that I held in the coffee. You have to strain your eyes a little, but you see the temperature of almost 80 degrees Celsius. And now you see how the coffee cools down at the original speed. It starts out quite fast. It is now 78 degrees, goes down to 77.8 and 77.7.

Then I used time lapse because it is really boring to watch coffee cool down for more than an hour. After about 70 minutes, it is a bit below 30 degrees.

However, you can also recognize something else, and to do so, you have to scroll back briefly. It is actually a nice exponential function produced by the results.

You see that you can do this with simple household objects and in principle in every classroom too. 

Slide 8

Let’s do another experiment; let’s roll a die. We roll it 10 times, 100 times, 1,000 times, and ask ourselves whether a 6 will always turn up in the results. What do you think?

Slide 9

I actually rolled a die. Certainly it was luck that when I rolled the die ten times, all numbers turned up, including the 6. You probably would have expected that for one hundred rolls. Would you also have expected the 2 to be relatively far behind? Presumably not. However, this expectation evidently didn't matter to chance.

Slide 10

What you see here is now the results for 1,000 rolls. A tally sheet no longer made sense in this case, but we see that these numbers are closer together. In particular, the 2 noticeably caught up.

As you know, the 6 plays a key role in many games, but you sometimes get the impression that you roll it less often. No, in this example that’s not the case at all.

Slide 11

It was apparently equally difficult to roll a 6 or any other number. Agreed?

Have we thus “proven” that all numbers between 1 and 6 turn up at approximately the same frequency when a die is rolled? Not really. However, the following assumption is reasonable:

If you roll a non-manipulated die n times, each of the events 1, 2, 3, 4, 5, 6 should occur approximately 𝑛/6  times when n is large.

Rolling a die is a LAPLACE experiment. In such an experiment, all results occur equally often in the long run.

Slide 12

This is Pierre-Simon de Laplace, a mathematician who was influential in the late 18th century and the early 19th century. In particular, he made important contributions to stochastics.

Slide 13

The “law of large numbers” applies to Laplace experiments: If you conduct a random experiment a very large number of times, then the relative frequency becomes closer to the theoretical probability (and we now simply use this term intuitively).

In simple terms: We need a very large number of experiments – as with our rolling a die – to be reasonably certain that a supposition is true.

Slide 14

Let’s take a look back. We have

  • elected a class representative
  • conducted a survey on how students get to school
  • observed the safety of means of transportation
  • observed the traffic at an intersection
  • watched coffee get cold and
  • (enthusiastically) rolled a die.

In the process, we have obtained very different data in very different ways and shown it differently. So one thing is certain: Statistics brings variety to our everyday lives, and not only at school.

Slide 15

This was a brief introduction to data collection. Thank you for listening and for your interest.

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